text:
\begin{document}

\title{Transience and Recurrence of Markov Processes \\ with Constrained Local Time}

 \author{Adam Barker}

\date{}

\maketitle

\begin{abstract}

 We study Markov processes conditioned so that their local time must grow slower than a prescribed function. Building upon recent work on Brownian motion with constrained local time in \cite{bb11,ks13}, we study transience and recurrence  for a broad class of Markov processes.

    In order to understand the   local time, we determine the distribution of a non-decreasing L\'evy process (the inverse local time) conditioned to remain above a given level which varies in time. We study a time-dependent     region, in contrast to previous works in which a process is conditioned to remain in a fixed region  (e.g.\   \cite{dw15,g09}), so we must  study boundary crossing probabilities for a family of curves, and thus obtain uniform asymptotics for such a family.

 Main results include necessary and sufficient conditions for transience or recurrence of the conditioned Markov process.   We will   explicitly determine the distribution of the  inverse local time for the conditioned process, and in the transient case, we explicitly determine the law of the conditioned Markov process. In the recurrent case, we characterise the ``entropic repulsion envelope'' via necessary and sufficient conditions.

\end{abstract}

\section{Introduction \& Background}

 We study the asymptotic behaviour of a Markov process whose local time is constrained to grow slower than $f$, an     increasing function.

The (right-continuous) inverse of the local time process is a subordinator (a non-decreasing L\'evy process)
, so our study of the behaviour of the local time process  is effectively equivalent to studying a  subordinator conditioned to grow faster than the inverse function $f^{-1}$.

This work is hence related to a number of works on stochastic processes conditioned to remain in a certain fixed region, such as cones in \cite{dw15,g09},
 and Weyl chambers in \cite{dw10,ks10}.   We highlight the fact that our subordinator is conditioned to remain in a region which varies in time, whereas the aforementioned works consider fixed regions, as appears to be the case for all works prior~to~\cite{ks13}.

We emphasise that in constraining the local time of a Markov process, the extent to which our constraint affects the process varies over time, depending on the past behaviour of the process.     This   is a ``weak'' constraint, in constrast to ``strong'' constraints such as conditioning a process to avoid a point, where the constraint does not change (see  e.g.\ \cite{b92,bbc03,c96}).    When our conditioned Markov process is recurrent, our constraint    varies over all time, whereas when our conditioned Markov process is transient, the constraint varies for only a finite window of time. So our   results, especially   in the recurrent case,  offer a significant contrast to many  prior works with a ``strong'' constraint.

Many works, e.g.\   \cite{aks12,bl16,dw14,kl13,l13,m16,pw01},
   consider a time-dependent region, and the passage time out of this region is studied.   In this paper, we study the  boundary crossing probability  for  a   family of curves  and study the time at which this crossing occurs (the function $f^{-1}$ forms our boundary of interest). We study these asymptotics,  uniformly, among a family of curves   in Lemma \mbox{\ref{lemma2}} (in contrast to prior non-uniform asymptotic results),   and   consider the deeper problem of determining the law of a subordinator conditioned to remain in this time-dependent region.

Our studies are also similar, in spirit, to  various other works on Brownian motion \cite{ks17,rvy06,p75},
   L\'evy processes \cite{b93,bbc03,c96,p13,yyy09},
    and more general diffusions \cite{sv09,rvy07} with restricted path behaviour.

  Specifically, this work is motivated by previous works on Brownian motion with constrained local time, such as \cite{bb11}, in which a 1-dimensional Brownian motion is conditioned so the local time at zero, $(L_t)_{t\geq0}$, satisfies $L_t\leq f(t)$ for all $t\geq0$, for a given function $f$, and a sufficient condition for transience of the conditioned Brownian motion is found.

  In \cite{ks13}, it is shown that the condition  is necessary and sufficient for transience of the conditioned Brownian motion, and the law of the conditioned inverse local time process is explicitly determined in both the transient and recurrent cases. In the transient case, an explicit formulation for the conditioned Brownian motion is found, and in the recurrent case the ``entropic repulsion envelope'' is found.

  This paper builds upon \cite{ks13} in particular, providing analogous results for a much broader class of processes than Brownian motion, with some mild regularity conditions.  It was conjectured in \cite[Remark 9]{ks13} that such analogous results hold when the inverse local time process has L\'evy measure with regularly varying tail, which we confirm in this paper. We   extend beyond     this conjecture   by including a much  more general setting, see Definition~\mbox{\ref{case2}}.

Constraining local time from above  imposes weak repellence on a Markov process.   Many such processes   are studied in works related to polymer physics, see e.g.\ \cite{bhp18,chp12,hk01,hk01a,w80}.

Particularly important is the  transition
between a localised phase, where the polymer remains close to a point, and a
delocalised phase, where it moves away.

The goal is often to understand
when the transition occurs as underlying model parameters vary, as in e.g.\ \cite{b02,bhp18,chp12,gk97,hkw11}.
 This motivates our study of   transience and recurrence of  Markov   processeses, transience and recurrence corresponding to  delocalised and localised phases, respectively.

 Now we  provide a brief exposition of the main result, before introducing some key definitions.

\p{Main Result} Starting with a recurrent Markov process, we constrain its local time $(L_t)_{t\geq0}$ so that $L_t\leq f(t)$ for all $t$. The following necessary and sufficient condition tells us if the constraint is strong enough to change the process to become transient: it is transient   if

\b{equation}
\label{criterion}
 \int_1^\infty     f(x)  \Pi(dx)     <\infty,
\e{equation}

and the process remains recurrent   otherwise, where $\Pi(dx)$ denotes the L\'evy measure of the inverse local time subordinator.

Our criterion $\l(\ref{criterion}\r)$ can also be understood in terms of the rate of growth of the inverse local time $(X_s)_{s\geq0}$ as $s\to\infty$, since it is known \cite[Theorem III.13]{b98} that
\[
\int_1^\infty     f(x)  \Pi(dx)     <\infty   \iff      \lim_{s\to\infty} \frac{X_s}{f^{-1}(s)} =0, \textup{ almost surely. }
\]

So the boundary choice of $f$, at which the conditioned process changes from recurrent to transient, coincides with the boundary at which $X_s$  grows to infinity faster or slower than $f^{-1}(s)$.

  The remainder of the paper is structured as follows: Section \mbox{\ref{keydefns}} provides key definitions; Section \hbox{\ref{mainresults}} outlines the statements of  the main results and the conditions under which they hold, including the necessary and sufficient conditions for transience/recurrence, the   distribution of the conditioned process, and the characterisation of the entropic repulsion envelope; Section  \hbox{\ref{mainresultsproofs}} contains the proofs of the main results; Sections  \hbox{\ref{lemma1proof}},  \hbox{\ref{lemma2proof}}, and \hbox{\ref{l6proof}} contain the proofs of 3 key lemmas required for the main results; Section  \hbox{\ref{lemmasproofs}} contains the proofs of the remaining auxiliary lemmas.

\section{Key Definitions}
\label{keydefns} We shall provide some definitions,  following   conventions   of \cite[Chapter IV]{b98}.
 \begin{definition} The local time $(L_t^x)_{t\geq0}$ of a Markov process $M$ at  $x$ is defined by
 \[
 L_t^x :=  \lim_{\eps\to0   }   \f{1}{2\eps}   \int_0^t   \bbm{1}_{ \{  |X_s -x| <\eps    \}   }ds.
 \]

 \end{definition}

 Without any loss of generality, we shall study only the local time of a Markov process at the origin, and denote $L_t:= L_t^0$.

  A subordinator is defined to be a non-decreasing real-valued stochastic process with stationary independent increments, started from 0.

 The right continuous inverse local time, defined by $X_t:= \inf\{ s>0 :  L_s > t   \}$,   is a subordinator.     The jumps of $(X_t)_{t\geq0}$ correspond to excursions of $(M_t)_{t\geq0}$ away from zero.

  The Laplace exponent $\Phi$ of a subordinator $X$ is defined by   $ e^{-\Phi(\lambda)} =   \mathbb{E}[  e^{- \lambda X_1}  ] $,    $\lambda\geq0$. By the   L\'evy-Khintchine formula \cite[p72]{b98},   $\Phi$ can   be written
 \begin{equation*}
 \label{lk}   \Phi(\lambda) = \ \text{d} \lambda +  \int_0^\infty  (1- e^{-\lambda x} ) \Pi(dx) ,
 \end{equation*}
 where d is the linear drift, and $\Pi$ is the L\'evy measure, which determines the size and rate of the jumps   of $X$, and satisfies $\int_0^\infty (1\wedge x)\Pi(dx)<\infty$.
We refer to  \cite{b98}
 for  background on subordinators.

Next, we define some important classes of functions with which we shall work.

   \b{definition}[Regular Variation and Related Properties]  \  \\
 \begin{enumerate}[(i)]
   \item  \label{rv}
         A   function $h:\bb{R}\to \bb{R}$ is regularly varying at $\infty$ with index $\alpha\in\bb{R}$ if    for all $\lambda>0$,  $\lim_{t\to\infty} h(\lambda t)/ h(t)   = \lambda^{\alpha} $. We refer to  \mbox{\cite{bgt89}} for background on regular variation.
  \\

\item \label{sv}  A   function $L:\bb{R}\to \bb{R}$ is slowly varying at $\infty$ if $\lim_{t\to\infty} L(\lambda t)/ L(t)   =1$ for each $\lambda>0$.  A  function $h$, regularly varying at $\infty$ of index $\alpha$, can always be written as $h(x) = x^\alpha L(x)$, where $L$ is slowly varying at $\infty$, see \mbox{\cite{bgt89}}.
\\

  \item \label{matus} The lower index, $\beta(h)$, of a function $h:\bb{R}\to \bb{R}$ is  the infimum of $\beta\in\mathbb{R}$ for which there exists $     C  > 0$  so that     for all  $    \Lambda  > 1$,    $  h(\lambda x)/  h(x)         \geq      (1+o(1))  C  \lambda^\beta$,  uniformly in   $ \lambda\in [1,\Lambda] $, as $x\to\infty$,~see~\cite[p68]{bgt89}.
\\

\item \label{crv}    A function $h$ is   CRV at $\infty$ if    $   \lim_{\lambda\to1} \lim_{t\to \infty }   h(\lambda t)/   h(t)       =1$.  The class of CRV functions  lies between ``extended regularly varying'' functions and   $\cal{O}$-regularly varying functions. See \cite{d98} for   details.
\\

 \item   \label{orv}      A function $h:\bb{R}\to \bb{R}$ is    $\cal{O}$-regularly varying at $\infty$ if   for each $\lambda>0$, it satisfies   both  $   \limsup_{ t\to \infty }        h(\lambda t) /   h(t)     <\infty$ and   $   \liminf_{ t\to \infty }   \   h(\lambda t) /   h(t)        >0$.   See \cite{d98} for further details.
\end{enumerate}
\e{definition}

\section{Statements of Main Results} \label{mainresults}    We aim to constrain the local time so that $L_t \leq f(t)$ for all $t\geq0$, where $f:[0,\infty)\to(0,\infty)$ is  increasing, $f(0)\in(0,1)$,     and $\lim_{t\to\infty}f(t)=\infty$. This work concerns  the behaviour  of our process as $t\to\infty$,   which is unaffected  by the  condition on $f(0)$. Before stating our main results, we   define the regularity conditions under which these~results~hold.

 \subsection{Regularity Conditions}  We shall  impose regularity  conditions on  the function  $f$, its   inverse function $g:=f^{-1}$ (extended so that for $x\in[0,f(0))$, $g(x)=0$), and the tail    $\ov(x):=\Pi(x,\infty)$ in two main cases of interest. Our conditions are imposed on the inverse local time subordinator rather than directly on the Markov process.

  Now let us define the  main cases of interest for our results:
\b{definition}[Case (i)]  \label{case1}     We impose on our subordinator that the   tail    $\ov(x)=\Pi(x,\infty)$ is regularly varying at $\infty$ with index $-\alpha\in(-1,0)$, so  $\ov(x)=x^{-\alpha} L(x)$ for $L$ slowly varying at $\infty$. We further impose   there exist $B,N>0$  such that the function  $x\mapsto  x^N L(x) $ is non-decreasing on   $(B,\infty)$.

\n We impose that   $f(0)\in(0,1)$,    $\lim_{t\to\infty} f(t) = \infty$, $f$ is  differentiable,  $tf'(t)  \ov(t) $ decreases to 0 as $t\to\infty$ (so $f$ is increasing), the inverse $g:=f^{-1}$ satisfies $\lim_{t\to\infty} g(t+\eps)/g(t)=1$ for all $\eps>0$,  and there exists some value $\beta>  (1+2\alpha)/(2 \alpha + \alpha^2) $ such that
  \b{equation} \label{condt}
  \lim_{t\to\infty}    t \ov\l(    \f{g(t)}{  \log(t)^\beta  } \r)    =0.
  \e{equation}
   \e{definition}

\b{remark}
Case (\hbox{\hyperref[case1]{i}}) includes stable subordinators, and subordinators whose L\'evy measure has similarly well-behaved tail asymptotics. Thus the set of Markov processes corresponding to  case    (\hbox{\hyperref[case1]{i}})  includes   Bessel processes, stable L\'evy processes of index $\alpha\in(1,2)$, and other Markov processes with similarly well-behaved asymptotics.   Case (\hbox{\hyperref[case2]{ii}}) corresponds to a much broader class~of~processes.

\e{remark}

   \b{definition}[Case (ia)]  \label{case1a} Under the assumptions of case (\hbox{\hyperref[case1]{i}}),     define   ``case  (ia)'' by     imposing   $f$, $f'$ are \mbox{\hyperref[orv]{$\cal{O}$-regularly varying}} at $\infty$,         the densities

    $ f_t(x) dx:=  \pp(X_t\in dx)  $ and  $u(x)dx := \Pi(dx)$ exist, and          there exist  constants $a,x_0\in(0,\infty)$,    such that for all $t\in(0,\infty)$ and  $x\geq  g(t) +  x_0$, where $g=f^{-1}$,
        \b{equation}   \label{l6cond}         f_t(x)    \leq      A  t  u(x).
    \e{equation}

  \e{definition}

    \b{remark} If $\ov$ is \mbox{\hyperref[rv]{regularly varying}} at $\infty$ and the density $f_t$   exists, then     $\l(\ref{l6cond}\r)$
    holds  for each fixed $t$ and $x>x(t)$, where $x(t)$ may depend on $t$            (see e.g.\     \cite[Theorem 1]{y94}). Here we further impose a bound on    $x(t)$, so that $\l(\ref{l6cond}\r)$ holds  uniformly    among sufficiently many  $x$ and $t$ for us to prove Theorem \mbox{\ref{entrrep}}.  For a stable subordinator of index $\alpha\in (0,1)$,  the density $f_t$ exists (see \cite[p227]{b98}) and $\l(\ref{l6cond}\r)$
    holds  (see Corollary \mbox{\ref{entrrepstable}}), so  case (\hbox{\hyperref[case1a]{ia}}) includes stable subordinators.
    \e{remark}

\b{definition}[Case (ii)]  \label{case2}  We impose on our subordinator that the   tail function    $\ov(x)=\Pi(x,\infty)$ is   CRV   at $\infty$, and has lower   index  $\beta(\ov)>-1$.

\n  We impose  that  $f(0)  \in (0,1)$,    for  $f$   increasing,  and that there exists $\eps> 0$ such that for $g := f^{-1}$,
   \b{equation}    \label{cond2}
    \lim_{t\to\infty}    t ^{1+\eps} \ov\l(   g(t)  \r)    =0.
    \e{equation}
  \e{definition}

       \b{remark} In Definition  \mbox{\ref{case2}}, we impose    $\beta(\ov)>-1$, which is equivalent to imposing that the function $ \int_0^x   \ov(y) dy$ has positive increase as $x\to\infty$. This has many equivalent formulations   \cite[Ex. III.7]{b98}, \cite[Section 2.1]{bgt89}, and appears naturally in a range of contexts \cite[p2]{b18}, \cite[p87]{b98}.
 \e{remark}

 \n  Now let us introduce some notation required to formulate our results.

      Recall that   $f:[0,\infty)\to(0,\infty)$ is  increasing,   $f(0)\in(0,1)$, and $g:=f^{-1}$ is the inverse of $f$, where  we take $g(x)=0$ for $x\in[0,f(0))$. The event $ \cal{O}_{u}$ corresponds to bounding the inverse local time until time $u$ (or equivalently, bounding the local time until time $g(u)$). We will study the asymptotics of $\pp(\cal{O}_u)$ as $u\to\infty$, and those of the integral $\Phi(s)$ of this probability.
 \b{align}
  \label{o}
  \cal{O}_{u} \    &:=    \{   X_s \geq   g(s),   \forall \  0\leq s\leq u      \}  ,    \phantom{\Big(}
  \\
\label{Phi}
  \Phi(s) \   &:= \int_0^s \pp(  \cal{O}_u )du     .              \phantom{\Big(}

   \intertext{We also study the event $ \cal{O}_{u}$ for the process  $X^{(0,a)}$ with truncated L\'evy measure $\Pi(dx)\mathbbm{1}_{x\in(0,a)}$,  }
 \label{truncoo}

    \cal{O}_{u,X^{(0,a)} }   &:= \l\{     X_s^{(0,a)}     \geq g(s), \forall \   0 \leq s \leq u   \r\}.

   \intertext{The time of our subordinator's first jump of size larger than $x$ is denoted by}

 \label{D}
  \Delta_1^x  \  &:=\inf\l\{ t\geq 0 : X_t - X_{t-} >x \r\}.       \phantom{\Big(}

     \intertext{In Corollary \mbox{\ref{transientcorollary}} and Proposition \mbox{\ref{lemma5}}, we determine that   $I(f)<\infty$ is a necessary and sufficient condition for transience of the conditioned process,~where}

     \label{}
   I(f)    &:=    \int_1^\infty   f(x)    \Pi(dx). \phantom{\Big(}

   \intertext{\b{remark} The necessary and sufficient condition    $I(f)  <    \infty$ arises naturally in a number of contexts, including rate of growth of subordinators \cite[Theorem III.13]{b98} and spectrally negative L\'evy processes \cite[Theorem 3]{p08}.
   \e{remark}
   }

    \intertext{For $h<t$, by the stationary independent increments property, the event $ \{X_t>g(t) | X_h =y\} $ is equivalent to $\{X_{t-h} > g(t) -y\} = \{X_{t-h} > g((t-h)+h) -y\}$.   This new   boundary  for $X$ to cross is given by $g_{y}^{h}(\cdot)$, with $ \cal{O}_u^{g_{y}^{h}} $, $   \Phi_y^h(s)$    corresponding to $ \cal{O}_u$, $\Phi(s)$.}

  \label{gyh}
   g_{y}^{h}(t)  &:= g(t+h)-y   ,     \phantom{\Big(}
     \\
       \label{ogyh}
  \cal{O}_{u}^{g_{y}^{h}}   &  :=   \{   X_s \geq   g_{y}^{h}(s),   \forall \  0\leq s\leq u      \}      ,            \phantom{\Big(}
  \\
  \label{phiyh}
  \Phi_y^h(s)    &:=\int_0^s   \pp(\cal{O}_{u}^{g_{y}^{h}})du     .            \phantom{\Big(}
     \end{align}

     \n    The functions $\rho(\cdot)$, $\rho_y^h(\cdot)$  are   error terms  in the   upcoming ODEs $\l(\ref{ODE1}\r)$~and~$\l(\ref{ODE2}\r)$.    \b{align}

       \label{rho}
  \rho(t) &:=\f{  \pp\l( \cal{O}_t\r) }{ \Phi(t)}    - \ov(g(t))     ,        \phantom{\Big(}
\\
 \label{rhoyh}
  \rho_y^h(t) &:=   \f{  \pp( \cal{O}_{t}^{g_{y}^{h}}) }{ \Phi_y^h(t)}    - \ov(g_{y}^{h}(t)).       \phantom{\Big(}
\end{align}

  \n     The law of our conditioned process will be found via the Doob h-transform. Recalling the notation (\ref{o}) and (\ref{ogyh}), for the measure $\mathbb{Q}(\cdot):=\lim_{t\to\infty} \mathbb{P}(\cdot | \cal{O}_t) $,
 \begin{align}
 \mathbb{Q}(X_h\in dy)  : =      \lim_{t\to\infty}  \mathbb{P}\l(    X_h \in dy    |   \cal{O}_t     \r) &=  \lim_{t\to\infty}  \mathbb{P}\l(    X_h \in dy ; \cal{O}_h   |   \cal{O}_t     \r) \nonumber

\\
 &=   \lim_{t\to\infty}    \f{  \mathbb{P}\l(   \cal{O}_t    |  X_h \in dy ; \cal{O}_h     \r)     \mathbb{P}\l( X_h \in dy ; \cal{O}_h   \r)   }{  \mathbb{P}(\cal{O}_t)    }    \nonumber
\\
 &=     \mathbb{P}\l( X_h \in dy ; \cal{O}_h    \r)     \lim_{t\to\infty}    \f{  \mathbb{P}(   \cal{O}_{t-h}^{g_{y}^{h}}        )      }{  \mathbb{P}(\cal{O}_t)    }   .      \label{doobh}
\end{align}
\n   In order to understand the   behaviour of  $X$ under $\mathbb{Q}$, we   study  the probabilities $ \mathbb{P}(     \cal{O}_{t}^{g_{y}^{h}}       )$ and $ \mathbb{P}(   \cal{O}_t       )$ as $t\to\infty$.   Lemma  \hbox{\ref{lemma1}},  proven in Section \hbox{\ref{lemma1proof}},   relates the asymptotics of $\pp(\cal{O}_t)$ to  $\Phi(t)$,  and   $ \mathbb{P}(     \cal{O}_{t}^{g_{y}^{h}}       )$ to $\Phi_y^h(t)$.
 We   obtain the~ODEs
\b{align}
\label{ODE1}
      \pp(\cal{O}_t )   &=   \f{d}{dt} \Phi(t)   =    \l(   \ov(g(t)) + \rho(t) \r)  \Phi(t) ,
      \\
      \label{ODE2}
      \pp(\cal{O}_{t}^{g_{y}^{h}} )   &=  \f{d}{dt}   \Phi_{y}^h (t)   =    \l(   \ov(g_{y}^{h}(t)) + \rho_y^h(t) \r)  \Phi_y^h(t) .

\intertext{These ODEs are easily solved, yielding}

\label{soln1}
\Phi(t) &=     \Phi(1) \exp\l(       \int_1^t  \ov(g(s))ds +        \int_1^t  \rho(s)ds        \r) ,
\\
\label{soln2}
\Phi_y^h(t) &=     \Phi_y^h(t_0(y)) \exp\l(       \int_{t_0(y)}^t  \ov(g_{y}^{h}(s))ds +        \int_{t_0(y)}^t  \rho_y^h(s)ds        \r) .
\end{align}
The error terms  $\rho(\cdot)$ and  $\rho_y^h(\cdot)$ are later shown to be integrable (the latter uniformly in $y$ and $h$), which is key for determining the distribution of our conditioned process. The required bound for  $\rho(\cdot)$ is given  in Remark \mbox{\ref{yh0}}, and we provide a uniform bound for $\rho_y^h(\cdot)$   in Lemma \hbox{\ref{lemma2}},  proven in Section    \hbox{\ref{lemma2proof}}.

\subsection{Results in the \textit{I}(\textit{f})  $<\infty$ Case} \label{transientsection}
We will see that when $I(f)<\infty$, our conditioned Markov process is transient. Theorem \mbox{\ref{transientthm}} finds the distribution of the conditioned inverse local time subordinator in this case.

 \b{theorem} \label{transientthm}   With assumptions on $f$ and $\ov$  in case (\hbox{\hyperref[case1]{i}}) or case (\hbox{\hyperref[case2]{ii}}), if $I(f)  <\infty$,
  \n then the limit measure $\mathbb{Q}(\cdot)$ exists   on the space  of  c\`adl\`ag (right continuous paths with left limits) paths on $[0,\infty)$, and   for all    $h>0$, $y>g(h)$,
  \[
  \mathbb{Q}(X_h \in dy)= \f{  \Phi_y^h(\infty) }{  \Phi(\infty)  }     \pp\l(     X_h \in dy ;\oo_h    \r) ,
  \]
   where  $\Phi(\infty)<\infty$, $\Phi_y^h(\infty) <\infty$.    Define the random variable $\frak{C}_t$ on $[0,t]$~by
 \b{equation} \label{frakC}
 \pp\l( \mathfrak{C}_t   \in ds\r) :=       \f{   \pp(\oo_s)   }{ \Phi(t)   } ds    ,
 \e{equation}
and define the weak limit $\frak{C}:= \lim_{t\to\infty}\frak{C}_t  $, which exists since $\Phi(\infty)<\infty$. Then   for all $h\geq 0$, $\mathbb{Q}(X_h <\infty)= \pp( \frak{C}>h)$.
  \e{theorem}

    \b{remark} \label{transientremark}  Since $\mathbb{Q}(X_h <\infty)= \pp( \frak{C}>h)$,  the   inverse local time $X$  jumps to infinity at a stopping time (for $X$'s filtration) with the same distribution as $\frak{C}$. Hence the Markov process $M$ never returns to 0 after time $X_\frak{C-}= \lim_{s\uparrow \frak{C}} X_s$.
  \e{remark}

 We shall now     describe, in the sense of weak convergence, the limiting behaviour of the Markov process $M$ until its first excursion longer than  $g(t)$, and when this excursion occurs.  Describing the limiting behaviour until the first excursion  longer than  $g(t)$, as $t\to\infty$, in fact gives full knowledge of how $M$ behaves under $\mathbb{Q}$ until the time of its final, infinite excursion. We   verify in Corollary  \hbox{\ref{transientcorollary}} that $M$ is transient, so never returns to 0 after the start of this final excursion.   Remark \mbox{\ref{panti}} considers the behaviour after this.

  \b{corollary}   \label{transientcorollary}        In cases (\hbox{\hyperref[case1]{i}}) and (\hbox{\hyperref[case2]{ii}}),      the process $(M,X)$ under the law $\pp(\cdot|\cal{O}_t)$ converges weakly as $t\to\infty$ to $(M,X)$ under $\mathbb{Q}$. Under $\mathbb{Q}$, $M$     behaves as   $ M$   conditioned on   $ \{   L_s    \leq    f(s),  \forall     s    \leq    X_\frak{C}     \}$         until   time   $ X_{\frak{C}-} $,   then $X$ jumps to infinity at time $\frak{C}$,  $  \lim_{t\to\infty}   \pp( \D^{g(t)}   \in   ds  |   \cal{O}_t ) =   \mathbb{P}(  \frak{C} \in ds  )$, so   $M$ is transient,  never visiting 0~after~time~$X_{\frak{C}-}$.
  \e{corollary}

 \b{remark}  \label{panti}
While the last excursion of the Markov process $M$ is not   dealt with explicitly, the behaviour of $M$ from time $X_{\frak{C}-}$ onwards  should be the same as that of $M$ conditioned to avoid zero.  When $M$ is a L\'evy process,  the behaviour of the process conditioned to avoid zero is well understood,  see   \cite[Theorem 8]{p13}.
 There is some technical difficulty in applying results from \cite{p13} to our final excursion.  The measure $\mathbb{Q}$ is constructed by conditioning until a deterministic time $t\to\infty$, but in \cite{p13}, the measure is constructed by conditioning until an independent exponential random time with   parameter $q\to0$. Equivalence of such deterministic and random limits is a separate~matter.
 \e{remark}

 \n Proposition  \hbox{\ref{transientlemma}}    formally captures the convergence in distribution discussed in Corollary \mbox{\ref{transientcorollary}}, and requires some understanding of excursion theory of Markov processes.  For  background, we direct the reader to \cite[Chapter IV]{b98}.
 \b{proposition} \label{transientlemma} For all fixed $b>a>h>0$,  $\cal{B} \in \cal{F}_h$, where $\cal{F}$ denotes the natural filtration of $X$, let $F_1 : E \to \mathbb{R}$ be a bounded continuous functional on the space $E$   of all possible excursions of $M$ away from 0. Defining the operator $\pi_h( (Z_u)_{u\geq0} ) :=   (Z_u)_{h\geq u \geq 0}$, let $F_1$ satisfy $F_1(\eps) = F_1 ( \pi_h(\eps))$ for all $\eps \in E$ ($F_1$ depends only on the excursion  in its first $h$   units of time). Then
     \b{align} \label{E}
 & \lim_{t\to\infty} \      \mathbb{E}  \l[   \ F_1(\pi_h(\eps))   \      \bbm{1}_{ \{ \pi_h(X) \in \cal{B}  \}  }       \bbm{1}_{ \{
\D^{g(t)} \in (a,b)  \}  }          \    \big|  \  \cal{O}_t  \ \r]
     \\
 &=         \int_{\nu \in \cal{B}}     \int_{u\in(a,b)}      \mathbb{E}  \l[       F_1(\pi_h(\eps ))       \big|      \pi_h(X) = \nu               \r]          \mathbb{Q}\l(     \pi_h(X) \in d\nu        ;    \frak{C} \in du  \r)                  \nonumber
\\
  &=       \mathbb{E}_\mathbb{Q}  \l[       F_1(\pi_h(\eps ))          \    \bbm{1}_{\{ \pi_h(X) \in \cal{B} \} }  \     \bbm{1}_{ \{      \frak{C} \in (a,b) \} } \r],                     \nonumber
  \end{align}
 where $\frak{C}$ is the    explosion time for $X$ under $\mathbb{Q}$, with distribution as   in (\ref{frakC}).
  \e{proposition}

\b{remark} All above  results in case (\hbox{\hyperref[case1]{i}}) with $I(f)<\infty$   in fact  hold   without the assumption that $f$ is differentiable or that $tf'(t) \ov(t) $  decreases to 0 as $t\to\infty$. We can also weaken the assumption $\l(\ref{condt}\r)$ by replacing $\beta$ by $1$, and all the results in case (\hbox{\hyperref[case1]{i}}) with $I(f)<\infty$ still hold, as does Lemma \mbox{\ref{lemma1}}.
\e{remark}

\subsection{Results in the  \textit{I}(\textit{f})    $=\infty$ Case}
    \label{recurrentsection}
     We now restrict our attention to case (\hbox{\hyperref[case1]{i}}).  We will see that when $I(f)=\infty$, our conditioned Markov process is recurrent. Theorem \mbox{\ref{recurrentthm}} finds the distribution of the conditioned inverse local time subordinator in this case.

 \b{theorem} \label{recurrentthm}  In case (\hbox{\hyperref[case1]{i}}), if $I(f)   =\infty$, then  the   law  $\mathbb{Q}(\cdot)= \lim_{t\to\infty} \pp(   \cdot  | \cal{O}_t  )$ exists (in the sense of  weak convergence), and the law of the inverse local time $X$ under $\mathbb{Q}$ satisfies for all $h>0$ and $y\geq g(h)$,
\b{align} \nonumber
 \mathbb{Q}\l( X_h \in dy    \r)     &=        \lim_{t\to\infty}    \f{  \mathbb{P}(   \cal{O}_{t-h}^{g_{y}^{h}}        )      }{  \mathbb{P}(\cal{O}_t)     }       \pp\l( X_h \in dy ; \cal{O}_h    \r)
  \\ \label{qqq}
      &=      q_h(y)   \pp\l( X_h \in dy ; \cal{O}_h    \r),
  \end{align}
   where  $q_h(y)$ is finite, non-decreasing in $y$,  and for $B>0$ as in Definition~\mbox{\ref{case1}},
  \[
q_h(y)     \hspace{-0.1cm}  =     \hspace{-0.1cm} \f{    \Phi_y^h(t_0(y))}{    \Phi(1)}      \hspace{-0.1cm} \lim_{t\to\infty}
      \exp\l(    \int_{  t_0(y)}^{t}       \hspace{-0.1cm}    \l(  \ov(g_{y}^{h}(s) )    \hspace{-0.1cm}  +    \hspace{-0.1cm}    \rho_y^h(s)   \r)   ds
 -    \hspace{-0.2cm}   \int_1^{ t   }    \hspace{-0.2cm}   \l(  \ov(g(s))  \hspace{-0.1cm}  +  \hspace{-0.1cm}  \rho(s)   \r) \hspace{-0.1cm} ds  \hspace{-0.05cm}     \r)  \hspace{-0.1cm}  ,
 \]
 \b{equation}
     \label{t0y}
    t_0(y) :=  f(Ay)\vee  f(1 + 2/A),   \qquad A>3\vee (B-1).      \phantom{\Big(}
    \e{equation}
   \e{theorem}

\n   Proposition  \hbox{\ref{lemma5}} verifies that when $I(f)=\infty$, $M$ is recurrent at 0 under  $\mathbb{Q}(\cdot)$, as the inverse local time never hits infinity at a finite time, $\mathbb{Q}$-almost surely.

  \b{proposition} \label{lemma5}   \label{Mweakconv}    In   case (\hbox{\hyperref[case1]{i}}), if $I(f) =\infty$, then  for each $h>0$,
  \[
  \mathbb{Q}\l( X_h \in (g(h),\infty)    \r)        =1.
  \]
 \e{proposition}

 \n Now let us describe the entropic repulsion envelope in Theorem \mbox{\ref{entrrep}}.

     \b{definition}   \label{entdefn} A non-decreasing function $w$, with $\lim_{h\to\infty}w(h)  =  \infty$, is in the entropic repulsion envelope $R_g$ (for the function $g=f^{-1}$) if
  \b{equation}
  \label{entdef}
  \lim_{h\to\infty}  \mathbb{Q}\l(  X_h \geq   w(h) g(h)    \r)   =1.
  \e{equation}
  \e{definition}

\b{theorem} \label{entrrep} In case (\hbox{\hyperref[case1a]{ia}}), a necessary and sufficient condition for non-decreasing  $w$,  for which $\lim_{h\to\infty}w(h)=\infty$, to be in $R_g$, for   $g=f^{-1}$  is
  \[
   w\in R_g \iff  \lim_{h\to\infty}   \int_h^{  f( w(h) g(h)   )  }     \ov(g(s)) ds =0.
  \]
  \e{theorem}
 \n As a result, one can    verify that the entropic repulsion envelope is always non-empty in case (\hbox{\hyperref[case1a]{ia}}),  by finding suitable  $w$.  We illustrate the generality of Theorem \mbox{\ref{entrrep}} via  the following corollary,   expanding upon \cite[Theorem 4]{ks13}.

  \b{corollary} \label{entrrepstable} In case (\hbox{\hyperref[case1]{i}}), with $f,f'$    \mbox{\hyperref[orv]{$\cal{O}$-regularly varying}}   at $\infty$, for a stable subordinator of index $\alpha\in(0,1)$,
  a necessary and sufficient condition for non-decreasing  $w$,   with $\lim_{h\to\infty}w(h)=\infty$, to be in  $R_g$, for   $g=f^{-1}$  is
  \[
   w\in R_g \iff  \lim_{h\to\infty}   \int_h^{  f( w(h) g(h) )  }   g(s)^{-\alpha} ds =0.
  \]
  \e{corollary}

\section{Proofs of Main Results} \label{mainresultsproofs}
This section contains the proofs of the results stated in Section \mbox{\ref{mainresults}}. First we state Lemmas \mbox{\ref{lemma1}} and \mbox{\ref{lemma2}},  proven in Sections \mbox{\ref{lemma1proof}} and \mbox{\ref{lemma2proof}}, respectively.
 \begin{comment}
       The law of our conditioned process is found via the Doob h-transform. Recall the notation (\ref{o}) and (\ref{ogyh}). For the measure $\mathbb{Q}(\cdot):=\lim_{t\to\infty} \mathbb{P}(\cdot | \cal{O}_t) $,
 \begin{align}
 \mathbb{Q}(X_h\in dy)  :\hspace{-0.1cm}&=      \lim_{t\to\infty}  \mathbb{P}\l(    X_h \in dy    |   \cal{O}_t     \r) \nonumber
\\
 &=  \lim_{t\to\infty}  \mathbb{P}\l(    X_h \in dy ; \cal{O}_h   |   \cal{O}_t     \r)  \nonumber
\\
 &=   \lim_{t\to\infty}    \f{  \mathbb{P}\l(   \cal{O}_t    |  X_h \in dy ; \cal{O}_h     \r)     \mathbb{P}\l( X_h \in dy ; \cal{O}_h   \r)   }{  \mathbb{P}(\cal{O}_t)    }    \nonumber
\\
 &=     \mathbb{P}\l( X_h \in dy ; \cal{O}_h    \r)     \lim_{t\to\infty}    \f{  \mathbb{P}(   \cal{O}_{t-h}^{g_{y}^{h}}        )      }{  \mathbb{P}(\cal{O}_t)    }   .      \label{doobh}
\end{align}
\n   So in order to understand the   behaviour of  $X$ under $\mathbb{Q}$, we can   study the asymptotics of the probabilities $ \mathbb{P}(     \cal{O}_{t}^{g_{y}^{h}}       )$ and $ \mathbb{P}(   \cal{O}_t       )$ as $t\to\infty$.
 This is done through Lemmas  \hbox{\ref{lemma1}} and  \hbox{\ref{lemma2}}, which are proven in Sections \hbox{\ref{lemma1proof}} and \mbox{\ref{lemma2proof}}.
 \end{comment}

  \b{lemma} \label{lemma1}   In cases (\hbox{\hyperref[case1]{i}}) and (\hbox{\hyperref[case2]{ii}}),  as $t\to\infty$,
   \b{equation} \label{l1}
   \pp\l( \oo_{t}   \r) =  \l(  \ov(g(t))+\rho(t)\r) \Phi(t)     = ( 1+ o(1)) \  \ov(g(t)) \Phi(t).
   \e{equation}
   \e{lemma}

   \b{definition} In this paper we   use the following asymptotic notation:

 $f(x) \sim g(x) $ as $x\to \infty$ if $\lim_{x\to \infty} f(x)/g(x) =1$.

   $f(x) \lesssim g(x) $ if  there exists $C\in(0,\infty)$ such that for all $x$, $f(x) \leq C g(x)$.

 \n   Moreover, we write   $f(x) \gtrsim g(x) $   if $g(x)\lesssim f(x)$,  and

 $f(x) \asymp g(x) $ if both $f(x) \gtrsim g(x) $ and $f(x) \lesssim g(x) $.

   \e{definition}

  \b{lemma} \label{lemma2}  In cases (\hbox{\hyperref[case1]{i}}) and (\hbox{\hyperref[case2]{ii}}),  there exist $u(t)$ with $\lim_{t\to\infty} u(t)    =   0$ and $\eps>0$,  such that uniformly for all    $A>3$, $h>0$, $y>g(h)$, and  $ t> t_0(y)$,
  \b{equation} \label{1}
  \rho_y^h(t)     \lesssim  \f{1}{   t \log(t)^{1+\eps} }    \l( 1 + \f{1}{f(y)-h} \r) ,
  \e{equation}
  \b{equation}\label{2}
    \rho_y^h(t)   \leq         u(t)\ov(g(t))   \l( 1 + \f{1}{f(y)-h} \r) .
  \e{equation}
   \e{lemma}

  \b{remark} \label{yh0} The inequality $\l(\ref{1}\r)$ also holds when $y=h=0$ (recall $f(0)>0$).  The proof is omitted - computations are simpler without dependence on $y$, $h$. This   implies   $\int_{t_0}^\infty   \rho(s)ds<\infty$, so by $\l(\ref{soln1}\r)$, uniformly as $t\to\infty$,
  \b{equation}
  \label{phiasymp}
  \Phi(t)   =   \Phi(1) \exp     \l(  \int_{1}^t (\ov(g(s))  +    \rho(s))ds       \r)    \asymp     \exp   \l(   \int_{1}^t \ov(g(s))ds     \r)  .
  \e{equation}
  \e{remark}

 \subsection{Proofs in the \textit{I(f)} $<\infty$ Case }

  \subsubsection{Proof of Theorem  \hbox{\ref{transientthm}}}
  \b{proof}[Proof of Theorem  \hbox{\ref{transientthm}}] First, let us verify that   $\Phi(\infty)<\infty$. Recalling that $g  :=  f^{-1}$, we have
\b{equation}   \label{If}
 I(f)
  \hspace{-0.05cm}=       \hspace{-0.1cm}    \int_1^\infty     \hspace{-0.25cm}    f(x) \Pi(dx)
  \hspace{-0.05cm}=       \hspace{-0.1cm}    \int_1^\infty  \hspace{-0.25cm}   \int_0^{f(x)}      \hspace{-0.25cm}   dy \Pi(dx)
  \hspace{-0.05cm}=     \hspace{-0.1cm}    \int_{0}^{\infty}   \hspace{-0.1cm}    \int_{1\vee g(y)}^\infty      \hspace{-0.1cm}    \Pi(dx) dy
  \hspace{-0.05cm}=     \hspace{-0.1cm}   \int_{0}^\infty  \ov(1\vee g(y))dy    .
\e{equation}
Now, recall from $\l(\ref{soln1}\r)$ that
\[
\Phi(\infty) =    \Phi(1)    \exp \l(    \int_{1}^\infty    \ov(g(s))   ds     +       \int_{1}^\infty    \rho(s)   ds        \r).
\]
By Lemma  \hbox{\ref{lemma1}},  as $s\to\infty$,   $\rho(s) =o(   \ov(g(s)) )$.   Then by $\l(\ref{If}\r)$, since $I(f)<\infty$,
\[
   \int_{1}^\infty    \ov(g(s))   ds     +       \int_{1}^\infty    \rho(s)   ds
 \overset{\ref{lemma1}}\lesssim       \int_{1}^\infty    \ov(g(s))   ds    \overset{\l(\ref{If} \r)}<  \infty ,
 \]

  so $\Phi(\infty)<\infty$. In particular, note that
\begin{equation}  \label{IfPhi}
   I(f) < \infty    \iff     \Phi(\infty)<\infty.
\end{equation}
To show $\Phi_y^h(\infty)<\infty$, recall by $\l(\ref{soln2}\r)$, for  $t_0(y)$ as defined in $\l(\ref{t0y}\r)$,
  \[
  \Phi_y^h(\infty) =    \Phi_y^h(t_0(y))    \exp \l(    \int_{t_0(y)}^\infty    \ov(g_{y}^{h}(s))   ds     +       \int_{t_0(y)}^\infty   \rho_y^h(s)   ds        \r).
  \]
Now, observe that for each fixed $y,h>0$, $g(s)\sim g_{y}^{h}(s)$ as $s \to\infty$, by  $\l(\ref{gyh}\r)$ and Definition \mbox{\ref{case1}}, so  $\ov(g(s)) \sim \ov(g_{y}^{h}(s))$ as $s\to\infty$, since $\ov$ is  \mbox{\hyperref[crv]{CRV}} at $\infty$. Now,  applying  $\l( \ref{2} \r)$ and $\l(\ref{If}\r)$, since $y$ and $h$ are fixed and $y>g(h)$ implies $f(y)-h>0$,
  \[
      \int_{t_0(y)}^\infty   \ov(g_{y}^{h}(s))   ds      +       \int_{t_0(y)}^\infty    \rho_y^h(s)   ds     \overset{   \l( \ref{2} \r)   }{\lesssim}        \l(   1 + \f{1}{f(y)-h} \r) \int_{t_0(y)}^\infty   \ov(g(s))   ds     \overset{\l(\ref{If}\r)}<    \infty,
  \]

\n    so $\Phi_y^h(\infty)<\infty$. By $\l(\ref{doobh}\r)$ and Lemma \mbox{\ref{lemma1}},  since $\ov(g(t)) \sim \ov(g_{y}^{h}(t-h))$ as $t\to\infty$,
 \b{align}
 \nonumber
\mathbb{Q}(X_h\in dy)    &\overset{  \l(\ref{doobh}\r)}=  \pp\l(    X_h \in dy      ;  \cal{O}_h    \r)     \lim_{t\to\infty}     \f{ \pp\big(   \oo_{t-h}^{g_{y}^{h}}          \big) }{   \pp\l(  \oo_{t}  \r)    }
\\

  &\overset{  \hspace{1.9pt}   \ref{lemma1}     \hspace{1.9pt}    }=      \pp\l(    X_h \in dy      ; \cal{O}_h    \r)     \lim_{t\to\infty}     \f{\ov(g_{y}^{h}(t-h)) \Phi_y^h(t-h) }{ \ov(g(t)) \Phi(t)   }

  \label{PQ}

 \overset{\phantom{   \l(\ref{doobh}\r)  }}=      \pp\l(    X_h \in dy      ; \cal{O}_h    \r)      \f{  \Phi_y^h(\infty) }{  \Phi(\infty)   } .
  \end{align}

  \n Now we show
  \(
     \mathbb{Q}(X_h     <          \infty)      \hspace{-2pt}       =   \hspace{-2pt}      \pp\l( \frak{C}   \hspace{-1pt}        >    \hspace{-1pt}      h   \r)
  \). Recall the notation $\Phi_y^h$ in $\l(\ref{phiyh}\r)$. By~$\l(\ref{PQ}\r)$,
 \b{align*}
\mathbb{Q}(X_h <\infty) &=     \int_{g(h)}^\infty  \mathbb{Q}(X_h \in dy)  =   \int_{g(h)}^\infty     \f{\Phi_y^h(\infty)    }{  \Phi(\infty)   }      \pp\l(    X_h \in dy  ; \cal{O}_h        \r)
 \\
  &=     \f{1}{\Phi(\infty)}  \int_{g(h)}^\infty    \int_0^\infty \pp\l(   \oo_{v}^{g_{y}^{h}}    \r) dv       \pp\l(    X_h \in dy         ; \cal{O}_h \r)    \\
  &= \f{1}{\Phi(\infty)}      \int_0^\infty    \int_{g(h)}^\infty    \pp\l( \oo_v^{g_{y}^{h}}    \r) \pp\l(   X_h\in dy   ; \cal{O}_h \r)  dv    .
 \intertext{Now,  $  \pp\big( \oo_v^{g_{y}^{h}}    \big) \pp\l(   X_h\in dy   ; \cal{O}_h \r) = \pp\l( \oo_{v+h}  ;  X_h\in dy   \r) $ by $\l(\ref{ogyh}\r)$. Then      by  the definition $\l(\ref{o}\r)$ of $\cal{O}_{v+h}$,  }

 \mathbb{Q}(X_h <\infty)
 &\overset{\phantom{(\ref{o})}}=      \f{1}{\Phi(\infty)}      \int_0^\infty    \int_{g(h)}^\infty  \hspace{-5pt}   \pp\l( \oo_{v+h}  ;  X_h\in dy   \r)  dv

&&\hspace{-65pt}\overset{\phantom{\ref{o}}}=     \f{1}{\Phi(\infty)}      \int_0^\infty   \hspace{-5pt}     \pp\l( \oo_{v+h}  ;  X_h > g(h)   \r)  dv\\
      &\overset{(\ref{o})}=     \f{1}{\Phi(\infty)}      \int_0^\infty        \pp\l( \oo_{v+h}      \r)  dv

       &&\hspace{-65pt}\overset{\phantom{\ref{o}}}=     \f{1}{\Phi(\infty)}      \int_h^\infty    \hspace{-5pt}    \pp\l( \oo_{u}     \r)  du =: \pp\l( \frak{C}   >h   \r)  .
  \end{align*}
\e{proof}

  \subsubsection{Proof of Corollary  \hbox{\ref{transientcorollary}}}

  \b{proof}[Proof of Corollary  \hbox{\ref{transientcorollary}}]
   Define  $Z^t:=    ((X_u)_{\D^{g(t)}\geq u\geq0} , \D^{g(t)}), t\geq0 $. We will show   $Z^t$ under $\pp( \cdot |   \cal{O}_t)$ converges (weakly) as $t\to\infty$  to  $Z:= \l(    (X_u)_{ \frak{C} \geq u \geq 0} , \frak{C}   \r)$, under $\mathbb{Q}$ with the desired law. This is sufficient for the desired weak convergence of   $X$ and   $ \D^{g(t)}$. The desired  weak convergence of $M$ until a last infinite excursion at time $\frak{C}$ then follows   from Proposition  \hbox{\ref{transientlemma}}.

  \n For $y > g(x)$, $ t > b > a > x >0$, with $a,b,x,y$   fixed, and an event $\cal{B}_X \in \cal{F}_{x}$, where $(\cal{F}_u)_{u\geq0}$ is the natural filtration of $X$, such that $\cal{B}_X\subseteq \cal{O}_x$, consider
  \b{equation}    \label{Jt}
    \lim_{t\to\infty}    \pp\l(       X_x \in dy   ;  \cal{B}_X ;     \D^{g(t)} \in (a,b) | \cal{O}_{t}   \r).
  \e{equation}
  Given $\Delta_1^{g(t)} >a>x$, we can replace $ \{ X_x \in dy \},   \cal{B}_X$ by     corresponding events  $ \{ X_x^{(0,g(t))} \in dy \} , \cal{B}_{X^{(0,g(t))}}$  for the   process $X^{(0,g(t))}$ with L\'evy measure  restricted to $(0,g(t))$, i.e.\    no jumps  larger than $g(t)$.  If $\D^{g(t)} \in ds$, then $X_s  >  g(t)$, so   $\cal{O}_t$ is fully  attained by time $s$, so   $\cal{O}_t$  can be replaced by $\cal{O}_s$,~and
  \b{align*}
\l(\ref{Jt}\r) &=   \lim_{t\to\infty}    \f{1}{\pp(\oo_t )}   \int_a^b         \pp\l(     X_x^{(0,g(t))}  \in dy  ;  \cal{B}_{X^{(0,g(t))}} ;     \D^{g(t)} \in ds ; \cal{O}_{t}       \r)
\\
  &=   \lim_{t\to\infty}    \f{1}{\pp(\oo_t )}   \int_a^b         \pp\l(    X_x^{(0,g(t))}  \in dy    ;  \cal{B}_{X^{(0,g(t))}} ;     \D^{g(t)} \in ds ; \cal{O}_{s}       \r) .

  \intertext{Recall the definition $\l(\ref{truncoo}\r)$. Given $\D^{g(t)} \in ds$, we can replace $\cal{O}_s $ by $\cal{O}_{s,X^{(0,g(t))}}$, where  $\cal{O}_{s,X^{(0,g(t))}}$  is independent~of~$\Delta_1^{g(t)}$, so}

 \l(\ref{Jt}\r)  &=    \lim_{t\to\infty}     \f{1}{\pp(\oo_t )}     \int_a^b         \pp\l(    X_x^{(0,g(t))} \in dy   ;  \cal{B}_{X^{(0,g(t))}} ;     \D^{g(t)} \in ds ; \cal{O}_{s,X^{(0,g(t))} }       \r)
\\
   &=   \lim_{t\to\infty}      \f{1}{\pp(\oo_t )}     \int_a^b         \pp\l(    X_x^{(0,g(t))} \in dy   ; \cal{B}_{X^{(0,g(t))}}  ; \cal{O}_{s,X^{(0,g(t))} }       \r)   \pp\l(   \D^{g(t)} \in ds\r)
\\
     &=    \lim_{t\to\infty}    \f{     \ov(g(t))    }{\pp(\oo_t )}     \int_a^b         \pp\l(    X_x^{(0,g(t))} \in dy   ;  \cal{B}_{X^{(0,g(t))}} ; \cal{O}_{s,X^{(0,g(t))} }       \r)   e^{- \ov(g(t)) s} ds.

 \intertext{Now, since $\lim_{t\to\infty} e^{- \ov(g(t)) s}   =1$, uniformly among $s\in(a,b)$, }

     \l(\ref{Jt}\r)      &=   \lim_{t\to\infty}       \f{     \ov(g(t))    }{\pp(\oo_t )}    \int_a^b         \pp\l(    X_x^{(0,g(t))} \in dy   ;  \cal{B}_{X^{(0,g(t))}}  ; \cal{O}_{s,X^{(0,g(t))} }       \r)  ds.

   \intertext{Applying   Lemma  \hbox{\ref{lemma1}}, and recalling    $\Phi(\infty)<\infty$  when $I(f)<\infty$ by $\l(\ref{IfPhi}\r)$, }

 \l(\ref{Jt}\r)   &=   \f{ 1}{   \Phi(\infty)     }    \lim_{t\to\infty}       \int_a^b         \pp\l(    X_x^{(0,g(t))} \in dy   ;  \cal{B}_{X^{(0,g(t))}}  ; \cal{O}_{s,X^{(0,g(t))} }       \r)  ds
\\
  &=      \f{1}{   \Phi(\infty)     }  \lim_{t\to\infty}      \int_a^b         \pp\l(    X_x^{(0,g(t))} \in dy   ;  \cal{B}_{X^{(0,g(t))}}   \big|  \cal{O}_{s,X^{(0,g(t))} }       \r)   \pp\l( \cal{O}_{s,X^{(0,g(t))} }  \r)  ds.

     \intertext{Now, $\lim_{t\to\infty}   \pp( \cal{O}_{s,X^{(0,g(t))} }  )  /    \pp\l( \cal{O}_{s }  \r)     =1$, uniformly among $s \in (a,b)$, so}

 \l(\ref{Jt}\r)   &=      \lim_{t\to\infty}          \int_a^b         \pp\l(    X_x^{(0,g(t))} \in dy   ;  \cal{B}_{X^{(0,g(t))}}   \big|  \cal{O}_{s,X^{(0,g(t))} }       \r)   \f{ \pp\l( \cal{O}_{s  }  \r)}{   \Phi(\infty)     }     ds,

  \intertext{and similarly $       \pp(    X_x^{(0,g(t))} \in dy   ;  \cal{B}_{X^{(0,g(t))}}   \big|  \cal{O}_{s,X^{(0,g(t))} }       )  \sim          \pp\l(    X_x \in dy   ;  \cal{B}_X   \big|  \cal{O}_{s }       \r) $ as $t\to\infty$, uniformly among $s\in(a,b)$. Then with $T_\infty:= \inf\{ t\geq0 : X_t-X_{t-}=\infty\}$,   by the definition of $\frak{C}$ in $\l(\ref{frakC}\r)$,      }

 \l(\ref{Jt}\r) &=           \int_a^b         \pp\l(    X_x  \in dy   ;  \cal{B}_X  \ \big| \ \cal{O}_{s }       \r)    \f{  \pp\l( \cal{O}_{s }  \r)   }{  \Phi(\infty)       } ds
 =:         \int_a^b         \pp\l(    X_x  \in dy   ;  \cal{B}_X  \ \big| \ \cal{O}_{s }       \r)   \bb{Q}\l(   \frak{C} \in ds\r)
\\
  &=          \int_a^b         \pp\l(    X_x  \in dy   ;  \cal{B}_X  \ \big| \ \cal{O}_{s };   T_\infty=s     \r)   \bb{Q}\l(   \frak{C} \in ds\r),

  \intertext{since $T_\infty=s$ does not affect the behaviour of $X$ up to time $x<s$. Then observing that $\cal{O}_s \cap \{T_\infty=s\} =\cal{O}_t \cap \{T_\infty=s\} =  $ for all $t\geq s$, it follows that  }

 \l(\ref{Jt}\r)  &=  \lim_{t\to\infty}         \int_a^b         \pp\l(    X_x  \in dy   ;  \cal{B}_X  \ \big| \ \cal{O}_{t };   T_\infty=s     \r)   \bb{Q}\l(   \frak{C} \in ds\r)
 \\
 &=         \int_a^b         \bb{Q}\l(    X_x  \in dy   ;  \cal{B}_X  \ \big| \    T_\infty=s     \r)   \bb{Q}\l(   \frak{C} \in ds\r)   =     \int_a^b         \bb{Q}\l(    X_x  \in dy   ;  \cal{B}_X  \ \big| \    \frak{C}=s     \r)   \bb{Q}\l(   \frak{C} \in ds\r)  ,
  \end{align*}
  since $T_\infty=\frak{C}$, $\bb{Q}$-almost surely, by Theorem \mbox{\ref{transientthm}}.
We conclude that for all $\cal{B} \in \cal{F}_a$ and $b>a>0$,
  \[
  \lim_{t\to\infty}     \pp(     Z^t   \in    \cal{B} \times (a,b)      |     \cal{O}_t)    =     \mathbb{Q} ( Z \in \cal{B} \times (a,b))     =     \mathbb{Q}\l(   (X_u)_{\frak{C}\geq u \geq 0} \in  \cal{B}_X  ;  \frak{C} \in (a,b)   \r)       ,
  \]
  so $Z^t $  converges weakly  to $Z$ as $t  \to  \infty$, and so $X$, $\Delta_1^{g(t)}$ also converge weakly,~as~required.

   \e{proof}

 \begin{comment}
 \p{Notation}

 \b{equation}
    \textup{$E_x$ denotes the space of possible excursions of length $x\geq0$}
\e{equation}

 \b{equation}
    \textup{$ \eps_s  $ denotes the excursion which begins at time $s$}
\e{equation}

 \b{equation}
    \textup{$\zeta( \eps_s  )$ denotes the length of an excursion $\eps_s \in E$}
\e{equation}

 \end{comment}

\subsubsection{Proof of Proposition  \hbox{\ref{transientlemma}}}
  \b{proof}[Proof of Proposition  \hbox{\ref{transientlemma}}]

Recall $\D^{g(t)}$ is the time of $X$'s first jump bigger than $g(t)$,  $\pi_h(X)$ is the sample  path of $X$ up to time $h$, $F_1$ is a  functional on the space of excursions from 0,  and  $\cal{B} \in \cal{F}_h$, where $(\cal{F}_u)_{u\geq0}$ is $X$'s natural filtration.  For  fixed $b>a>h>0 $,   disintegrating on the values   $\D^{g(t)},\pi_h(X)$,
  \b{equation} \label{E}
  \mathbb{E}\l[       \bbm{1}_{ \{ \pi_h(X) \in \cal{B}  \}  }    \    \bbm{1}_{ \{
\D^{g(t)} \in (a,b)  \}  }       F_1(\pi_h(\eps))     \    \big|  \  \cal{O}_t  \ \r]
  \e{equation}
  \[
  =  \hspace{-0.2cm} \underset{\nu\in\cal{B}}\int    \    \underset{ u \in(a,b)}\int  \hspace{-0.3cm}   \mathbb{E} \big[       F_1(\pi_h(\eps))      \big|      \cal{O}_t   ;  \pi_h(X) = \nu        ;    \D^{g(t)} =u     \big]      \pp\big(   \pi_h(X) \in d\nu        ;    \D^{g(t)} \in du    | \cal{O}_t   \big).
  \]

 Given a fixed path  $\pi_h(X)= \nu$, $\pi_h(\eps)$ depends only on $\nu$, so  $\pi_h(\eps)$ is  independent   of $\D^{g(t)}$ and $\cal{O}_t$.  Here the   excursion $\eps$ has length at most $g(t)$, so we may   replace $\pi_h(\eps) $ by $ \pi_h( \eps^{g(t)} ) $,  where    $\eps^{g(t)}   $    is an excursion sampled using the conditional excursion measure on the space $E^{g(t)}$  of excursions of length  at most $ g(t)$, so
  \b{align*}
  \l(\ref{E}\r)

  &=     \int_{\cal{B}}     \int_a^b       \mathbb{E} \l[      F_1(\pi_h(\eps^{g(t)} ))       \big|    \pi_h(X) = \nu          \r]    \pp\l(   \pi_h(X) \in d\nu        ;    \D^{g(t)} \in du    | \cal{O}_t   \r),
  \end{align*}

  Now, observe that
  \b{align*}
  \lim_{t\to\infty} \mathbb{E}  \l[  \hspace{0.01cm}     F_1(\pi_h(\eps^{g(t)} ))   \      \big|   \   \pi_h(X) = \nu  \     \r] &=  \mathbb{E}  \l[       F_1(\pi_h(\eps ))       \big|      \pi_h(X) = \nu               \r],
\\
     \lim_{t\to\infty}      \pp\l(      \pi_h(X) \in d\nu        ;    \D^{g(t)} \in du    |    \cal{O}_t   \r)
      &= \mathbb{Q}\l(     \pi_h(X) \in d\nu        ;    \frak{C} \in du  \r)            .
   \end{align*}
Disintegrating on the values of $\pi_h(X)$ and $\frak{C}$, we conclude, as required, that
  \b{align*}
  \lim_{t\to\infty}    \l(\ref{E}\r) &=         \int_{\cal{B}}     \int_a^b       \mathbb{E}  \l[       F_1(\pi_h(\eps ))       \big|      \pi_h(X) = \nu               \r]     \mathbb{Q}\l(     \pi_h(X) \in d\nu        ;    \frak{C} \in du  \r)
\\
  &=       \mathbb{E}  \l[       F_1(\pi_h(\eps ))         \    \bbm{1}_{\{ \pi_h(X) \in \cal{B} \} }  \     \bbm{1}_{ \{      \frak{C} \in (a,b) \} } \r].
  \end{align*}

  \e{proof}

  \subsection{Proofs in the \textit{I}(\textit{f}) $=\infty$ Case }

  \n  The next three proofs require Lemmas  \hbox{\ref{lemma4}} and \mbox{\ref{lowerboundforrho}}, both proven in Section  \hbox{\ref{lemmasproofs}}.

 \b{lemma} \label{lemma4}
In case  (\hbox{\hyperref[case1]{i}}),     for $t_0(y)$ as in~$\l(\ref{t0y}\r)$, uniformly in    $h  >  0, y  >  g(h)$,
 \b{equation} \label{l4}
\int_{t_0(y)}^\infty     \l(   \ov(g(s+h)-y) - \ov(g(s))     \r) ds     \lesssim   y f'(y) \ov(y)     .
 \e{equation}
 \e{lemma}

 \b{lemma} \label{lowerboundforrho} In case (\hbox{\hyperref[case1]{i}}), for  $\rho$ as  defined in $\l(\ref{rho}\r)$,  $\liminf_{t\to\infty} \rho(t) \geq0$.

 \e{lemma}

    \subsubsection{Proof of Theorem \mbox{\ref{recurrentthm}}}

  \b{proof}[Proof of Theorem  \hbox{\ref{recurrentthm}}]

     For each fixed $h>0$, $y>g(h)$,  we  will  now prove   that $    q_h(y):=      \lim_{t\to\infty}      \pp(       \cal{O}_{t-h}^{g_{y}^{h}}       ) / \pp(\cal{O}_t) < \infty$. For   each $h  >  0$, $y  >  g(h)$, note $g(t)\sim g_{y}^{h}(t-h)$ by Definition \mbox{\ref{case1}}. Hence   $\ov(g(t)) \sim \ov(g_{y}^{h}(t-h))$ as $t\to\infty$, since  $\ov$ is \mbox{\hyperref[crv]{CRV}} at $\infty$. By  Lemma~\hbox{\ref{lemma1}},
  \b{equation} \label{firstlim}
  \lim_{t\to\infty}  \f{\pp\big(       \cal{O}_{t-h}^{g_{y}^{h}}       \big) }{ \pp(\cal{O}_t)}      =   \lim_{t\to\infty}      \f{ \ov( g_{y}^{h}(t-h)) \Phi_y^h(t)   }{   \ov( g(t)) \Phi(t) }    =   \lim_{t\to\infty}          \f{  \Phi_y^h(t)   }{    \Phi(t) }.
  \e{equation}
Then by $\l(\ref{soln1}\r)$ and $\l(\ref{soln2}\r)$, for   $t_0(y)$ as defined in (\ref{t0y}),
  \b{align*}
\l( \ref{firstlim}\r)  \hspace{-0.05cm} &=    \hspace{-0.05cm}  \f{        \Phi_y^h(   t_0(y)  )    }{   \Phi(1)   }    \lim_{t\to\infty}     \exp\l(    \hspace{-0.05cm}  \int_{   t_0(y) }^{t}   \hspace{-0.2cm}     \l(  \ov(g_{y}^{h}(s))     \hspace{-0.05cm}  +  \hspace{-0.05cm}  \rho_y^h(s)   \r)   ds
 -  \hspace{-0.15cm}    \int_1^{ t   }  \hspace{-0.1cm}  \l(  \ov(g(s)) +   \rho(s)   \r) ds   \hspace{-0.05cm}    \r)   \hspace{-0.1cm} .

\intertext{By $\l(\ref{1}\r)$ in Lemma  \hbox{\ref{lemma2}},   the integral $  \int_{   t_0(y) }^{\infty}      \rho_y^h(s) ds    $ is uniformly bounded for all $h>0$, $y>g(h)$.
By Lemma \mbox{\ref{lowerboundforrho}}, it follows that     $- \int_1^\infty \rho(s) ds <\infty$,~so}

  \l(\ref{firstlim} \r) &\lesssim    \f{        \Phi_y^h(   t_0(y)  )    }{   \Phi(1)   }  \lim_{t\to\infty}   \exp\l(   \int_{   t_0(y) }^{t}      \ov(g_{y}^{h}(s))     ds
 -     \int_1^{ t   }  \ov(g(s))   ds     \r)
\\
  &\lesssim       \f{        \Phi_y^h(   t_0(y)  )    }{   \Phi(1)   }    \lim_{t\to\infty}     \exp\l(   \int_{   t_0(y) }^{t}   \l(   \ov(g_{y}^{h}(s))     -        \ov(g(s))   \r)  ds     \r).

\intertext{Applying Lemma  \hbox{\ref{lemma4}}, and recalling $ y f'(y)  \ov(y)$  decreases to zero as $y\to\infty$,}

   \l(\ref{firstlim} \r) &\lesssim   \f{        \Phi_y^h(   t_0(y)  )    }{   \Phi(1)   }  \exp\l(  y f'(y) \ov(y)  \r)  <\infty.
  \end{align*}
  Now, $q_h(y) :=    \lim_{t\to\infty}      \pp(       \cal{O}_{t-h}^{g_{y}^{h}}       ) / \pp(\cal{O}_t) $ is non-decreasing in $y$ since for all $y<y'$, $g_{y}^{h}(t)= g(t+h) -y > g(t+h) -y'  = g_{y',h}(t)$, and so   $ \pp(       \cal{O}_{t-h}^{g_{y}^{h}}       ) \leq     \pp(       \cal{O}_{t-h}^{g_{y',h}}       )$.  Finally, by  $\l(\ref{doobh}\r)$,
  \(
  \mathbb{Q}\l( X_h \in dy     \r)         =       \pp\l(X_h \in dy   ;  \cal{O}_h \r) q_h(y),
  \) as required.

  \e{proof}

  \b{proof}[Proof of Proposition  \hbox{\ref{lemma5}}] For   $h>0$,       $    \lim_{t\to\infty}  \pp\l(    X_h \in (g(h),\infty) | \cal{O}_t    \r)=1$.
    We  will prove by dominated convergence  \cite[Theorem 1.21]{k06} that     limits and integration can be exchanged   from $\l(\ref{intt1}\r)$ to $\l(\ref{intt2}\r)$,   so by $\l(\ref{doobh}\r)$,~for~all~$h>0$,
  \b{align}          \nonumber
 1=  \lim_{t\to\infty}   \pp\l(    X_h \in (g(h),\infty) | \cal{O}_t    \r)  &=
       \lim_{t\to\infty}     \int_{g(h) }^\infty     \pp\l(    X_h \in dy | \cal{O}_t    \r)
      \\
       &=         \label{intt1}
          \lim_{t\to\infty}     \int_{g(h) }^\infty     \f{  \pp(    \cal{O}_{t-h}^{g_{y}^{h}} )   }{   \pp(\cal{O}_t)  }    \pp\l(    X_h \in dy ; \cal{O}_h    \r)
  \\
    &=            \label{intt2}
          \int_{g(h) }^\infty     \lim_{t\to\infty}    \f{  \pp(    \cal{O}_{t-h}^{g_{y}^{h}} )   }{   \pp(\cal{O}_t)  }    \pp\l(    X_h \in dy ; \cal{O}_h    \r)
   \\
       &=      \label{exchange}
           \mathbb{Q}\l(     X_h \in (g(h),\infty)    \r)  ,
  \end{align}

  \n as required. For $A  >  3\vee   (B-1)$, we will  bound the integral over $(g(h),\infty)$ via:
  \b{equation}   \label{I1I2I3}
     \l[  \f{ g(t-h)}{ A}  ,  \infty  \r)  \cup      (g(h) ,  g(h+1)] \cup \l( g(h+1) , \f{ g(t-h)}{ A}  \r)
  =:           I_1 \cup I_2   \cup  I_3.
  \e{equation}

    \p{Proof for \mbox{\hyperref[I1I2I3]{$I_1$}}}  Since $y\in  I_1 $ if and only if $t \leq  f(Ay ) + h$,   by  Lemma  \hbox{\ref{lemma1}},
  \b{align*}
  \int_{I_1}        \f{  \pp(    \cal{O}_{t-h}^{g_{y}^{h}} )   }{   \pp(\cal{O}_t)  }    \pp\l(    X_h \in dy ; \cal{O}_h    \r)     \lesssim
  \int_{I_1}          \f{ 1 }{  \ov(g(t))   \Phi(t) }  \pp\l(    X_h \in dy     \r)
=  \f{       \pp\l(    X_h \geq      \f{g(t-h)}{A}        \r)     }{  \ov(g(t))   \Phi(t) }        .
  \end{align*}
  Now,  $I(f)=\infty$, so   $\lim_{t\to\infty} \Phi(t) = \infty$ by $\l(\ref{IfPhi}\r)$, and it suffices  to show that
  \b{equation} \label{markov}
  \limsup_{t\to\infty}  \f{    \pp\l(    X_h \geq      \f{g(t-h)}{A}     \r)    }{   \ov(g(t))  }
  \e{equation}
 is finite for each fixed $h>0$, as   the integal in $\l(\ref{intt1}\r)$ over the region $I_1$ tends to 0 as $t\to\infty$, so the dominated convergence theorem applies, trivially, on~$I_1$.

\n Recall $\l(\ref{D}\r)$. Observe  that   $\D^{g(t)}$ has exponential distribution of~rate~$\ov(g(t))$,~so

  \b{align}    \label{partitioning}     \nonumber
    \pp     \hspace{-0.05cm} \l(  \hspace{-0.1cm}  X_h \geq       \f{g(t-      \hspace{-0.05cm}  h)}{A}      \hspace{-0.05cm}   \r)  \hspace{-0.1cm} &=
      \pp     \hspace{-0.05cm} \l(         \hspace{-0.05cm}     X_h \geq        \f{g(t-      \hspace{-0.05cm}  h)}{A}      ;   \D^{g(t)} >     \hspace{-0.1cm}  h     \hspace{-0.05cm}   \r)  \hspace{-0.1cm}
      +   \hspace{-0.05cm} \pp       \hspace{-0.05cm}  \l(   \hspace{-0.15cm} X_h \geq        \f{g(t-      \hspace{-0.05cm}  h)}{A}     ;   \D^{g(t)}     \hspace{-0.15cm} \leq h    \hspace{-0.1cm} \r)
 \\\nonumber
  &\leq     \ \   \pp\l(    X_h^{(0,g(t))}  \geq    \f{g(t-h)}{A}        \r)    \ \   +       \pp\l(     \D^{g(t)} \leq h  \r)
 \\\nonumber
  &=    \ \  \pp\l(    X_h^{(0,g(t))}  \geq       \f{g(t-h)}{A}       \r)   \ \  +      1 -  e^{ -  h    \ov(g(t))}
  \\
  &\leq  \ \    \pp\l(    X_h^{(0,g(t))}\geq        \f{g(t-h)}{A}      \r)     \ \   +      h    \ov(g(t)) ,
  \end{align}
  where $X^{(0,g(t))}$ has   the same L\'evy measure  as $X$, but restricted to $(0,g(t))$, so $X^{(0,g(t))}$ has no jumps larger than $g(t)$.
By $\l(\ref{partitioning}\r)$ and Markov's inequality,
  \b{align*}
    \l(  \ref{markov}  \r)    -  h
   &\leq
       \limsup_{t\to\infty}  \f{    \pp\big(    X_h^{(0,g(t))}   \geq        \f{g(t-h)}{A}     \big)    }{   \ov(g(t))  }

      \lesssim          \limsup_{t\to\infty}     \f{  A   \mathbb{E}[   X_h^{(0,g(t))} ]}{     \ov(g(t)) g(t  -  h)      }

      =          \limsup_{t\to\infty}    \f{ A h \int_0^{g(t)}   x \Pi(dx)      }{   \ov(g(t))    g(t  -  h)    }   .

      \intertext{Now,   $ \int_0^{g(t)}   x \Pi(dx)     =   \int_{x=0}^{g(t)} \int_{y=0}^x dy    \Pi(dx)     =      \int_{y=0}^{g(t)}   \int_{x=y}^{g(t)} \Pi(dx) dy  \leq    \int_0^{g(t)}  \ov(y) dy     $. Then as $\ov$ is \mbox{\hyperref[rv]{regularly varying}} at $\infty$, by   Karamata's theorem \cite[Prop 1.5.8]{bgt89}, we deduce, as required for  dominated convergence  on $I_1$,~that~for~each $h>0$,  }

           \l(  \ref{markov}  \r)        &\lesssim       h     +  \limsup_{t\to\infty}   \f{     g(t) \ov(g(t))      }{     \ov(g(t)) g(t-h)   }
 <\infty.
 \end{align*}

    \p{Proof for \mbox{\hyperref[I1I2I3]{$I_2$}}}

By  Theorem \mbox{\ref{recurrentthm}},  $q_h(y) $ is non-decreasing in $y$, so by $\l(\ref{qqq}\r)$,  $ \pp(       \cal{O}_{t-h}^{g_{y}^{h}}       )$ is non-decreasing in $y$.  Now,    $ \lim_{t\to\infty}     \pp(    \cal{O}_{t-h}^{g_{ g(h+1),h}} )  /  \pp(\cal{O}_t)          = q_h(g(h+1))<\infty$ for each fixed $h$ by     Theorem \mbox{\ref{recurrentthm}}, and we conclude that
  \b{align*}
  \lim_{t\to\infty}   \int_{I_2}         \f{  \pp(    \cal{O}_{t-h}^{g_{y}^{h}} )   }{   \pp(\cal{O}_t)  }    \pp\l(    X_h \in dy ; \cal{O}_h    \r)          &\leq    \lim_{t\to\infty}       \f{  \pp(    \cal{O}_{t-h}^{g_{ g(h+1),h}} )   }{   \pp(\cal{O}_t)  }         \int_{I_2}       \pp\l(    X_h \in dy ; \cal{O}_h    \r)
\\
  &=   q_h(g(h+1))         \pp\l(    X_h \in I_2 ; \cal{O}_h    \r)  ,
  \end{align*}
which is finite for each   $h \hspace{-1.6pt} >  \hspace{-1.6pt}   0$, so   dominated~convergence applies~on~$I_2$.

    \p{Proof for \mbox{\hyperref[I1I2I3]{$I_3$}}}  By $\l(\ref{ODE2}\r)$ and Lemma \mbox{\ref{lemma1}}, for all large enough $t$,
 \b{align} \nonumber
 \f{  \pp(    \cal{O}_{t-h}^{g_{y}^{h}} )   }{   \pp(\cal{O}_t)  }
  &\leq 2      \f{    \l[   \ov(g_{y}^{h}(t-h))   +     \rho_y^h(t-h)    \r] \Phi_y^h(t-h)             }{     \ov(g(t))  \Phi(t)}    .
  \intertext{For $y\in I_3$, $y>g(h+1)$, so $f(y)-h> f(g(h+1))-h =1$,  and $ 1+ 1/(f(y)-h)   <2 $.       By $\l(\ref{2}\r)$,  as $\lim_{t\to\infty} u(t)=0$, for all large enough $t$ and for all $y \in I_3$,    }

\nonumber
  \f{  \pp(    \cal{O}_{t-h}^{g_{y}^{h}} )   }{   \pp(\cal{O}_t)  }    &\leq 2      \f{ \l( 1 +  u(t)  (1 + \f{1}{f(y)-h}  )  \r) \ov(g_{y}^{h}(t-h))     \Phi_y^h(t-h)             }{     \ov(g(t))  \Phi(t)}
  \\
  \nonumber
  \f{  \pp(    \cal{O}_{t-h}^{g_{y}^{h}} )   }{   \pp(\cal{O}_t)  }    &\leq 6      \f{     \ov(g_{y}^{h}(t-h))     \Phi_y^h(t-h)             }{     \ov(g(t))  \Phi(t)}       .

  \intertext{Now, $y<g(t-h)/A< g(t)/3$   for $y\in I_3$, so by $\l(\ref{gyh}\r)$, since $\ov$ is
  regularly varying at $\infty$,        $\ov(g_{y}^{h}(t-h))= \ov(g(t)- y)  \lesssim \ov(g(t))$, uniformly among $y\in I_3$.   So   for each fixed $h>0$, for all large enough $t$, uniformly in $y \in I_3$,    }

    \label{uno}
    \f{  \pp(    \cal{O}_{t-h}^{g_{y}^{h}} )   }{   \pp(\cal{O}_t)  }      &\lesssim        \f{    \Phi_y^h(t-h)    }{   \Phi(t)}.

 \intertext{Now, by $\l(\ref{soln2}\r)$, for $t_0(y)$ as defined in $\l(\ref{t0y}\r)$,}

\nonumber
\Phi_y^h(t-h) &= \Phi_y^h(t_0(y)) \exp\l(      \int_{t_0(y)}^{t-h}  \ov(g_{y}^{h}(s))ds  +     \int_{t_0(y)}^{t-h}   \rho_y^h(s)ds       \r).

\intertext{Applying    $\l(\ref{1}\r)$ and  recalling  that $ 1+ 1/(f(y)-h)   <2 $ for $y\in \mbox{\hyperref[I1I2I3]{$I_3$}}$,   the integral  $\int_{t_0(y)}^{\infty}   \rho_y^h(s)ds $ is uniformly bounded among $y$, so uniformly among $y\in I_3$,     }

\nonumber
 \Phi_y^h(t-h) &\lesssim

 \Phi_y^h(t_0(y)) \exp\l(     \int_{t_0(y)}^{t-h}  \ov(g_{y}^{h}(s))ds  \r)      .

 \intertext{By Lemma \mbox{\ref{lowerboundforrho}},    $ \liminf_{t\to\infty} \rho(t) \geq 0$,   so   $ \liminf_{t\to\infty} \int_1^t   \rho(s) ds > - \infty$, then~by~$\l(\ref{ODE1}\r)$,}

\nonumber
 \Phi(t) &=   \Phi(1) \exp\l(    \int_1^t \ov(g(s))ds  +     \int_1^t \rho(s)ds  \r)     \gtrsim   \exp\l(    \int_1^t \ov(g(s))ds    \r).

 \intertext{Since $y> g(h+1)$ in $I_3$, recalling $\l(\ref{t0y}\r)$,   $t_0(y)   \geq  f(Ay) >  f(y) \geq h+1 >1$,~so
     }

\nonumber
 \f{ \Phi_y^h(t-h) }{    \Phi(t)}    &\lesssim  \Phi_y^h(t_0(y)) \exp  \hspace{-0.05cm}   \l( \hspace{-0.05cm}    \int_{t_0(y)}^{t-h} \hspace{-0.05cm}    \ov(g_{y}^{h}(s))ds     - \hspace{-0.1cm}    \int_1^t   \hspace{-0.05cm}  \ov(g(s))ds      \r)
\\
\nonumber
&\leq       \Phi_y^h(t_0(y) )  \exp\l(      \int_{t_0(y)}^{t}   \l(\ov(g_{y}^{h}(s))   - \ov(g(s))    \r) ds          \r).

\intertext{Now, by Lemma  \hbox{\ref{lemma4}}, since $  y  f'(y) \ov(y) $ decreases to zero as $y\to\infty$,
\[
  \limsup_{t\to\infty}   \hspace{-1pt}   \l(       \sup_{y\in I_3}  \int_{t_0(y)}^{t}      \hspace{-2pt}   \l( \ov(g_{y}^{h}(s))    \hspace{-2pt}     -      \hspace{-2pt}  \ov(g(s))    \r)   \hspace{-2pt}     ds    \hspace{-2pt}    \r)    \hspace{-2pt}         \lesssim    \hspace{-1pt}    \limsup_{t\to\infty}   \hspace{-2pt}    \l( \sup_{y\in I_3}         y  f'(y) \ov(y)   \hspace{-2pt}   \r) <\infty  ,
\]
uniformly in $y>g(h)$. So    for each fixed $h>0$,  uniformly in $y\in I_3$,  by   $\l(\ref{phiyh}\r)$,     }

\\
    \lim_{t\to\infty}   \f{ \Phi_y^h(t-h) }{    \Phi(t)}
  &\lesssim    \Phi_y^h(t_0(y))  \leq       t_0(y).             \label{dos}
\end{align}

Now, it follows from $\l(\ref{uno}\r)$ and  $\l(\ref{dos}\r)$ that for each fixed $h>0$,
\b{equation}    \label{eqn}
       \lim_{t\to\infty}     \int_{ I_3}    \f{  \pp(    \cal{O}_{t-h}^{g_{y}^{h}} )   }{   \pp(\cal{O}_t)  }    \pp\l(    X_h \in dy ; \cal{O}_h \r)
 \lesssim
        \lim_{t\to\infty}     \int_{  I_3}       t_0(y)     \pp\l(    X_h \in dy ; \cal{O}_h \r).
\e{equation}

\b{align*}

\intertext{Note that for   $y\in \mbox{\hyperref[I1I2I3]{$I_3$}}$, $y<  g(t-h)/A<g(t)/A$. Then it follows that for all large enough $t$,   $t_0(y) = f(Ay) \vee f(1- 2/A)   <t $  ,     recalling the notation $\l(\ref{o}\r)$ and $\l(\ref{D}\r)$, by $\l(\ref{condt}\r)$,}

  \l(   \ref{eqn}\r)    &\leq           \lim_{t\to\infty}         t    \int_{ I_3}        \pp\Big(    X_h \in dy ; \cal{O}_h \Big)              \phantom{\bigg(}

 &&\hspace{-65pt}\leq         \lim_{t\to\infty}         t  \ \pp  \Big(  X_h \leq \f{g(t)}{A}   \Big)                 \phantom{\bigg(}

\\

&=        \lim_{t\to\infty}        t  \  \pp\Big(    X_h \leq \f{g(t)}{A}      ;      \Delta_1^{g(t)} > h  \Big)                 \phantom{\bigg(}

&&\hspace{-65pt}\leq        \lim_{t\to\infty}       t \  \pp\l(    \Delta_1^{g(t)} > h  \r)                   \phantom{\bigg(}

\\

&=         \lim_{t\to\infty}       t \l(1-e^{-h\ov(g(t))} \r)                 \phantom{\bigg(}

&&\hspace{-65pt}\leq         \lim_{t\to\infty}      ht  \ov(g(t))   \overset{\l(\ref{condt}\r)}=0 .
\end{align*}
Therefore     the dominated convergence theorem applies  on $I_3$, and the order of limits and integration can be swapped between $\l(\ref{intt1}\r)$ and $\l(\ref{intt2}\r)$,~as~required.

\nopagebreak
 \e{proof}

  \subsubsection{Proof of Theorem \mbox{\ref{entrrep}}}

 \n   Lemmas \mbox{\ref{l6}}, \mbox{\ref{qh}}, \mbox{\ref{bigjump}}, \hbox{\ref{1fyh}},    proven in Sections \mbox{\ref{l6proof}},\mbox{\ref{lemmasproofs}}, help prove Theorem \mbox{\ref{entrrep}}.

      \b{lemma}   \label{l6}      For each subordinator and $g=f^{-1}$   in case (\hbox{\hyperref[case1a]{ia}}),          there exist $K,h_0>0$ such that for all   $h> h_0$, with $\ov(y)  =  y^{-\alpha} L(y) $, uniformly in~$y  >  K$,
  \b{equation}
  \label{l6statement}
   \pp\l(   X_h \in g(h) dy ;   \cal{O}_h    \r)     \asymp     y^{-1-\alpha}    \f{L(g(h)y)}{L(g(h))}  \pp(\cal{O}_h) dy
         \e{equation}
    \e{lemma}

 \b{lemma} \label{qh}    For $\d>0$ small enough that $0<f(0)<f(\d)<1$, uniformly for all $h>0$ and $y>g(h+f(\d))$,
  \b{equation*} \label{qbound}
   q_h(y)  \asymp     \Phi_y^h(f(Ay))    \exp\l(       -       \int_{
   1}^{f(Ay)} \ov(g(s))   ds \r).
   \e{equation*}
  \e{lemma}

     \b{lemma} \label{bigjump}
   For a subordinator    in case (\hbox{\hyperref[case1a]{ia}}), and for a function $g$ which increases to $\infty$,    let $S_{\Delta_1^{g(h)}}$ denote the size of    its first jump of size greater than $g(h)$. Then there is $h_0>0$ such that uniformly for all  $h>h_0$ and $v>1$,
   \[
   \pp\l(   S_{\Delta_1^{g(h)}} \in g(h)dv  \r)      =     \f{ \Pi(g(h) dv)   }{  \ov(g(h))   }   \asymp    \f{  L(g(h)v) }{L(g(h))}  v^{-1-\alpha} dv.
   \]
  In particular there is $x_0\in (0,\infty)$ so that for all $x>x_0$, with $\Pi(dx)  =  u(x)dx$,
   \begin{equation} \label{2.10.3}
         u(x)   \asymp  x^{-1} \ov(x)    =  L(x) x^{-1-\alpha}    .
   \end{equation}
   \e{lemma}

  \b{lemma} \label{1fyh}  Recall the notation $\l(\ref{phiyh}\r),\l(\ref{t0y}\r)$. If  $h>0$,       $y>g(h)$, and $t\geq f(Ay)$,  for $A>3\vee (B-1)$,  then $\Phi_y^h(t) \geq  f(y)- h$.

  \e{lemma}

 \b{proof}[Proof of Theorem \ref{entrrep}]  As   $ \mathbb{Q}( X_h  \geq   g(h))  =  1$ for   $h  \geq  0$,~by~Def.\ \mbox{\ref{entdefn}},
 \b{equation}
 \label{wiff}
 w\in R_g       \iff  \lim_{h\to\infty}  \mathbb{Q}\l(  X_h \in (g(h),   w(h) g(h))    \r)   =0  .
 \e{equation}
   Now, by Theorem \mbox{\ref{recurrentthm}},
  \b{equation}    \label{integ}
  \lim_{h\to\infty}  \mathbb{Q}\l(  X_h \in (g(h),   w(h) g(h))    \r)   =    \lim_{h\to\infty} \int_{g(h)}^{w(h)g(h)}  q_h (y) \pp\l( X_h \in dy ; \cal{O}_h    \r) .
  \e{equation}
  We begin by showing that if  $\lim_{h\to\infty}   \int_h^{  f( w(h) g(h) )  }     \ov(g(s)) ds =0$, then $w\in R_g$.
   \p{Proof of Sufficient Condition}    Let $\lim_{h\to\infty}   \int_h^{  f( w(h) g(h) )  }     \ov(g(s)) ds =0$.    To show   $w\in R_g$, we will show  that the limit of the integral~$\l(\ref{integ}\r)$~is~zero on~each~of
   \[
     \label{R123}   [g(h),g(h+1)] \cup  [g(h+1),Kg(h)]   \cup    [g(h+1)\vee Kg(h),w(h)g(h)] =: R_1 \cup R_2 \cup R_3
     \]
       separately, where $K$ is the constant  in Lemma $\l(\ref{l6}\r)$.

 Note that if $g(h+1)> K g(h)$, then we need only consider  $R_1 \cup R_3$. Since $g$ is non-decreasing, we only need to consider the value $K$ if $K>1$.

  \p{Proof for $R_1$}  By Theorem \mbox{\ref{recurrentthm}}, $q_h(y)$ is non-decreasing in $y$, so
  \b{align}
  \label{h+1}
 &  \lim_{h\to\infty} \int_{g(h)}^{ g(h+1)}  q_h (y) \pp\l( X_h \in dy ; \cal{O}_h    \r)
\\
 \nonumber
   & \leq     \lim_{h\to\infty}    q_h(g(h+1))  \int_{g(h)}^{ g(h+1)}  \pp\l( X_h \in dy ; \cal{O}_h    \r)

  \leq      \lim_{h\to\infty} q_h(g(h+1))   \pp\l(  \cal{O}_h    \r) .

     \intertext{Applying Lemma \mbox{\ref{qh}}, then applying Lemma \mbox{\ref{lemma1}},}
     \nonumber
     \l(\ref{h+1}\r) &\lesssim     \lim_{h\to\infty}   \Phi_{g(h+1)}^h(f(Ag(h+1)))   e^{      -       \int_{1}^{f(Ag(h+1))} \ov(g(s))   ds      }        \pp\l(  \cal{O}_h    \r)
     \\
     \nonumber
     &=   \lim_{h\to\infty}   \Phi_{g(h+1)}^h(f(Ag(h+1)))  e^{      -       \int_{1}^{f(Ag(h+1))} \ov(g(s))   ds      }         \Phi(h)      \ov(g(h))   .

     \intertext{Now, $ \Phi_{g(h+1)}^h(f(Ag(h+1)))  \leq   f(Ag(h+1)) \lesssim    h$, and   $f(Ag(h+1)) \geq   h$, as  $f=g^{-1}$ is \mbox{\hyperref[orv]{$\cal{O}$-regularly varying}}, increasing, and $A>1$.  By $\l(\ref{phiasymp}\r)$ and $\l(\ref{condt}\r)$,}

     \nonumber
          \l(\ref{h+1}\r) &\lesssim  \lim_{h\to\infty}    h e^{   -       \int_{1}^{f(Ag(h+1))} \ov(g(s))   ds    }     \Phi(h)       \ov(g(h))
  \\
  \nonumber
  &\leq       \lim_{h\to\infty}    h e^{   -       \int_{1}^{h} \ov(g(s))   ds    }     \Phi(h)       \ov(g(h))
          \overset{\l(\ref{phiasymp}\r)}\lesssim    \lim_{h\to\infty}   h  \ov(g(h))  \overset{\l(\ref{condt}\r)}=0.
     \end{align}

                      \p{Proof for    \mbox{\hyperref[R123]{$R_2$}}}  Recall that we only need to consider $R_2$ when $g(h+1)< Kg(h)$, in which case $K$ must satisfy $K>1$.  By Theorem \mbox{\ref{recurrentthm}}, $q_h(y)$ is non-decreasing in $y$, so
  \b{align}
 \label{limint}
   &  \lim_{h\to\infty} \int_{g(h+1)}^{Kg(h)}  q_h (y   ) \pp\l( X_h \in dy ; \cal{O}_h    \r)
\\
 \nonumber
  &\leq    \lim_{h\to\infty}  q_h (Kg(h) )   \int_{g(h+1)}^{Kg(h)}   \pp\l( X_h \in dy ; \cal{O}_h    \r)

\leq     \lim_{h\to\infty}  q_h (Kg(h) )   \pp\l( \cal{O}_h    \r).

\intertext{Applying Lemma \mbox{\ref{lemma1}}, then   Lemma \mbox{\ref{qh}} (note $g(h+f(\d))<g(h+1)< Kg(h)$), }

 \nonumber  \l( \ref{limint}\r)    &\overset{\ref{lemma1}}\leq     \lim_{h\to\infty}  q_h (Kg(h) )       \Phi(h)\ov(g(h)).
\\
  \nonumber
   &\overset{\ref{qh}}\lesssim      \lim_{h\to\infty}  \Phi_{Kg(h)}^h ( f(AKg(h))  )    e^{   - \int_1^{ f(AKg(h))   }    \ov(g(s))ds  }     \Phi(h)\ov(g(h)).

\intertext{Observe that $\Phi_{Kg(h)}^h(t)\leq t$  for all $t>0$, by $\l(\ref{phiyh}\r)$, and  $f(AKg(h)) \geq f(g(h))=h$ since $f$ increasing and $A,K>1$.      Moreover, as $f$ is  $\cal{O}$-regularly varying at $\infty$,  $f(AKg(h)) \lesssim f(g(h))=h$, as $h\to\infty$,  so}

\nonumber
  \l( \ref{limint}\r)   &\leq  \lim_{h\to\infty}       f(AKg(h))     \exp\l(    - \int_1^{  f(AKg(h))  }    \ov(g(s))ds  \r)     \Phi(h)\ov(g(h)).

\\

 \nonumber
  &\leq  \lim_{h\to\infty}     f(AKg(h))  \exp\l(    - \int_1^{  h }    \ov(g(s))ds  \r)     \Phi(h)\ov(g(h))
  \\
  \nonumber
  &\lesssim  \lim_{h\to\infty}   h   \exp\l(    - \int_1^{  h }    \ov(g(s))ds  \r)     \Phi(h)\ov(g(h)).

\end{align}
By $\l(\ref{phiasymp}\r)$,  $\Phi(h)  \asymp   \exp(  \int_1^h \ov(g(s))ds )$ as $h\to\infty$, so as $\lim_{h\to\infty} h \ov(g(h))  =0$ by $\l(\ref{condt}\r)$, we conclude  that  $\l(\ref{limint}\r)=0$, so the integral over $R_2$ is zero, and  thus
\b{equation}
\label{R1R2}
    \lim_{h\to\infty}     \int_{R_1\cup R_2}        q_h (y) \pp\l( X_h \in dy ; \cal{O}_h    \r)  = 0 .
 \e{equation}

\p{Proof for   \mbox{\hyperref[R123]{$R_3$}}}   Now we wish to show convergence to zero of
  \b{equation} \label{kw}
    \int_{g(h+1)\vee Kg(h)}^{w(h)g(h)}  q_h (y) \pp\l( X_h \in dy ; \cal{O}_h    \r)    =      \int_{\f{g(h+1)}{g(h)}\vee K }^{w(h) }  q_h (g(h)v) \pp\l( X_h \in g(h)dv ; \cal{O}_h    \r) .
  \e{equation}

 Applying Lemma \mbox{\ref{l6}}, then changing variables back to $y= g(h)v$,   recalling that $\ov(g(h))= g(h)^{-\alpha} L(g(h))$,    then applying Lemma \mbox{\ref{lemma1}}, as $h\to\infty$,
  \b{align}
\nonumber
\l(\ref{kw}\r) &\overset{\ref{l6}}\asymp     \int_{\f{g(h+1)}{g(h)}\vee K  }^{w(h) }  q_h (g(h)v)  v^{-1-\alpha}  \f{ L(g(h)v)}{L(g(h))}   \pp(\cal{O}_h)    dv

\\

\nonumber
  &\overset{\phantom{\ref{l6}}}=    \f{ \pp\l(\cal{O}_h\r) g(h)^{\alpha}  }{ L(g(h))   }    \int_{\f{g(h+1)}{g(h)}\vee K  }^{w(h) }  q_h (g(h)v)  g(h)^{-\alpha}   v^{-1-\alpha} L(g(h)v) dv

\\

\nonumber
  &\overset{\phantom{\ref{l6}}}=      \f{ \pp\l(\cal{O}_h\r)  }{ \ov(g(h))   }  \hspace{-2.5pt}   \int_{g(h+1)\vee  K g(h)}^{w(h)g(h) }      q_h (y)    y^{-1-\alpha} L(y) dy

  \overset{\ref{lemma1}}\sim     \Phi(h)  \hspace{-5pt}  \int_{g(h+1)\vee  Kg(h) }^{w(h)g(h) }      q_h (y)    y^{-1-\alpha} L(y) dy.

\intertext{Changing variables \hspace{-2pt}  ($ u     =   Ay$), applying Lemma \mbox{\ref{qbound}} and then the uniform convergence theorem \cite[Theorem 1.2.1]{bgt89},   as $L$ is \mbox{\hyperref[sv]{slowly varying}} at $   \infty       $ and  $\Phi_y^h(f(u))    \leq    f(u)$,    it follows that as $h  \to  \infty,$}

\nonumber
   \l(\ref{kw}\r) &\overset{\ref{qbound}}\asymp       \Phi(h)   \int_{g(h+1)\vee  Kg(h) }^{w(h) g(h)}   \Phi_y^h(f(Ay))    \exp\l(       -       \int_{1}^{f(Ay)} \ov(g(s))   ds \r)   y^{-1-\alpha}L(y) dy

\\
\nonumber
  &\overset{\phantom{\ref{qbound}}}\leq        \Phi(h)   \int_{ Kg(h) }^{w(h) g(h)}   \Phi_y^h(f(Ay))    \exp\l(       -       \int_{1}^{f(Ay)} \ov(g(s))   ds \r)   y^{-1-\alpha}L(y) dy

\\
\nonumber
  &\overset{\phantom{\ref{qbound}}}\lesssim       \Phi(h)   \int_{AKg(h) }^{Aw(h) g(h)}   \Phi_y^h(f(u))    \exp\l(       -       \int_{1}^{f(u)} \ov(g(s))   ds \r)   u^{-1-\alpha}L(u) du

\\

\nonumber
   &\overset{\phantom{\ref{qbound}}}\leq       \Phi(h)   \int_{AKg(h) }^{Aw(h) g(h)}  f(u)    \exp\l(       -       \int_{1}^{f(u)} \ov(g(s))   ds \r)   u^{-1-\alpha}L(u) du.

   \intertext{Since $A,K>1$,  we can split up the integral as follows, and we will deal with each term separately:}

\nonumber
   \l(\ref{kw}\r)  &\lesssim     \Phi(h)   \int_{g(h) }^{w(h) g(h)}  f(u)    \exp\l(       -       \int_{1}^{f(u)} \ov(g(s))   ds \r)   u^{-1-\alpha}L(u) du
\\
\nonumber
  &+    \Phi(h)   \int_{w(h)g(h) }^{Aw(h) g(h)}  f(u)    \exp\l(       -       \int_{1}^{f(u)} \ov(g(s))   ds \r)   u^{-1-\alpha}L(u) du
\\
\label{J12}
  &=: J_1(h) +   J_2(h).
  \end{align}

  \p{Proof for $J_2(h)$}  \hspace{-0.3cm}  As     $f(u)  \hspace{-0.025cm}   \geq   \hspace{-0.025cm}    f(w(h)g(h))  \hspace{-0.025cm}    \geq   \hspace{-0.025cm}   f(g(h))  \hspace{-0.025cm}   =  \hspace{-0.025cm}   h$ for $u  \hspace{-0.025cm}   \geq  \hspace{-0.025cm}    w(h)g(h)$,~by~$\l(\ref{phiasymp}\r)$,
 \b{align}
 \nonumber
  J_2(h) &\overset{\phantom{(\ref{phiasymp})}}=   \Phi(h)   \int_{w(h)g(h) }^{Aw(h) g(h)}  f(u)   e^{   -       \int_{1}^{f(u)} \ov(g(s))   ds  }     u^{-1-\alpha}L(u) du
 \\
 \nonumber
 &\overset{\phantom{(\ref{phiasymp})}}=       \Phi(h)   e^{     -       \int_{1}^{h} \ov(g(s))   ds }     \int_{w(h)g(h) }^{Aw(h) g(h)}  f(u)  e^{      -       \int_{h}^{f(u)} \ov(g(s))   ds  }    u^{-1-\alpha}L(u) du
 \\
 \nonumber
  &\overset{(\ref{phiasymp})}\lesssim   \hspace{-5pt} \int_{w(h)g(h) }^{Aw(h) g(h)}    \hspace{-5pt}  f(u)  e^{      -       \int_{h}^{f(u)} \ov(g(s))   ds  }    u^{-1-\alpha}L(u) du

 \overset{\phantom{(\ref{phiasymp})}}\leq  \hspace{-5pt}     \int_{w(h)g(h) }^{Aw(h) g(h)}    \hspace{-5pt} f(u)      u^{-1-\alpha}L(u) du.

  \intertext{Since $f$, $f'$ are \mbox{\hyperref[orv]{$\cal{O}$-regularly varying}} at $\infty$, one can verify that, uniformly for all sufficiently large $u$, $f(u)/u  \asymp  f'(u)$,  see   \cite[Prop 2.10.3]{bgt89}.  Recall  that in case (\hbox{\hyperref[case1]{i}}), $ f'(u)   u \ov(u)$ decreases to 0 as $u\to\infty$, so   as $h\to\infty$,}

\nonumber
    J_2(h) &\lesssim             \int_{w(h)g(h) }^{Aw(h) g(h)}  f'(u)      u^{-\alpha}L(u) du

  =     \int_{w(h)g(h) }^{Aw(h) g(h)}   u  f'(u)    \ov(u)  u^{-1}  du.
\\
\nonumber
  &\leq           w(h)g(h)  \      f'\big(w(h)g(h)\big)    \    \ov\big(w(h)g(h)\big)       \int_{w(h)g(h) }^{Aw(h) g(h)} u^{-1} du
\\
\label{j2}
  &= o(1) \times \int_{w(h)g(h) }^{Aw(h) g(h)} u^{-1} du = o(1) \times \log(A)= o(1),
  \end{align}
so $\lim_{h\to\infty} J_2 (h)  =0$, and $J_2(h)$ never contributes. Now we consider   $J_1(h)$.

  \p{Proof for \mbox{\hyperref[J12]{$J_1(h)$}}} First, changing variables from $s$ to $ v:=g(s)$ (so $s=f(v)$),
  \b{align}
\nonumber
  J_1(h) &=      \Phi(h)   \int_{g(h) }^{w(h) g(h)}  f(u)    \exp\l(       -       \int_{1}^{f(u)} \ov(g(s))   ds \r)   u^{-1-\alpha}L(u) du
\\
\nonumber
  &=     \Phi(h)   \int_{g(h) }^{w(h) g(h)}  f(u)    \exp\l(       -       \int_{g(1)}^{u} \ov(v)  f'(v) dv \r)   u^{-1-\alpha}L(u) du.

  \intertext{Recall $u^{-\alpha} L(u)   =   \ov(u)$, and $f(u)   \asymp     u  f'(u)$ uniformly as $u  \to  \infty$,  so as $h  \to  \infty$, }

\nonumber
    J_1(h) &\asymp     \Phi(h)   \int_{g(h) }^{w(h) g(h)}  f'(u)    \exp\l(       -       \int_{g(1)}^{u} \ov(v)  f'(v) dv \r)   u^{-\alpha}L(u) du
\\
\nonumber
  &=       \Phi(h)   \int_{g(h) }^{w(h) g(h)}  f'(u) \ov(u)    \exp\l(       -       \int_{g(1)}^{u} \ov(v)  f'(v) dv \r)   du.

 \intertext{Changing variables from $u$ to $z:= e^{-\int_{g(1)}^u \ov(v) f'(v) dv}$ and applying $\l(\ref{phiasymp}\r)$, it follows that as $h\to\infty$,   }

\nonumber
    J_1(h)   &\overset{\phantom{\l(\ref{phiasymp}\r)}}\asymp

  \Phi(h)  \l[   e^{-\int_{g(1)}^{g(h)}     \ov(v)f'(v) dv  }   -  e^{-\int_{g(1)}^{w(h)g(h)}   \ov(v)f'(v) dv  }        \r]
\\

  &\overset{\phantom{\l(\ref{phiasymp}\r)}}=       \Phi(h)  \l[   e^{-\int_{1}^{h}     \ov(g(s))ds  }   -  e^{-\int_{1}^{f(w(h)g(h))}   \ov(g(s))  ds }        \r]

     \label{j1}

     \overset{\l(\ref{phiasymp}\r)}\asymp         1-  e^{   -\int_{h}^{f(w(h)g(h))}   \ov(g(s))  ds }.
  \end{align}
    Thus by $\l(\ref{R1R2}\r)$, $\l(\ref{j2}\r)$ and $\l(\ref{j1}\r)$,    whenever   $\lim_{h\to\infty}\int_{h}^{f(w(h)g(h))}   \ov(g(s))  ds  =0$,  $w$ is in   the entropic repulsion envelope  $R_g$, as required for the sufficient condition.

 \n    Now we will prove that if $w\in R_g$, then $ \lim_{h\to\infty}   \int_h^{f(w(h)g(h))}   \ov(g(s)) ds =0$.

 \p{Proof of Necessary Condition}
 Let $w\in R_g$. Then by $\l(\ref{qqq}\r)$,
 \b{align}

 \nonumber
  0 &\overset{\phantom{\ref{l6}}}=   \lim_{h\to\infty}  \mathbb{Q}\l(  X_h \in (g(h),   w(h) g(h))    \r)

 \\
  \nonumber
  &
  \overset{\phantom{\ref{l6}}}=    \lim_{h\to\infty}      \int_{ g(h)}^{ w(h) g(h)}   q_h(  y)    \pp\l(    X_h \in     dy ; \cal{O}_h   \r)

  \overset{\phantom{\ref{l6}}}=    \lim_{h\to\infty}      \int_{ K  g(h) }^{ w(h)  g(h) }   q_h(  y)    \pp\l(    X_h \in    dy ; \cal{O}_h   \r)    ,

    \intertext{since the limit of  the integral over  $R_1 \cup R_2$ is always zero by $\l(\ref{R1R2}\r)$. Changing variables to $v= y / g(h) $, then applying Lemma \mbox{\ref{l6}},      }

      \nonumber
0     &\overset{\phantom{\ref{l6}}}=    \hspace{-4pt}    \lim_{h\to\infty}      \int_{ K    }^{ w(h)    }  \hspace{-9pt}   q_h(  g(h)v)    \pp\l(    X_h \hspace{-2pt} \in \hspace{-2pt}  g(h)dv  ; \cal{O}_h   \r)

 \asymp    \lim_{h\to\infty}      \pp(\cal{O}_h)     \hspace{-4pt}    \int_{K   }^{w(h)  }  \hspace{-9pt}  q_h( g(h)v )        v^{-1-\alpha}  \f{ L(g(h)v) }{ L(g(h))}  dv.
   \intertext{Changing variables   to $y=g(h)v$ and recallling that $\ov(g(h))= g(h)^{-\alpha} L(g(h))$, then applying   Lemmas \mbox{\ref{lemma1}}, \mbox{\ref{qh}}, and \mbox{\ref{1fyh}},    }

\nonumber
0 &\overset{\phantom{\ref{1fyh}}}=      \lim_{h\to\infty}    \f{ \pp(\cal{O}_h) }{ \ov(g(h))}    \int_{K g(h)}^{w(h)g(h)}     \hspace{-7pt} q_h( y )       y^{-1-\alpha}  L(y)      dy

  \overset{\hspace{2pt} \ref{lemma1} \hspace{2pt} }=      \lim_{h\to\infty}      \Phi(h)       \int_{K g(h)}^{w(h)g(h)}      \hspace{-5pt} q_h( y )               y^{-1-\alpha}  L(y)   dy
\\
\nonumber
 &\overset{  \hspace{2pt}  \ref{qh}  \hspace{2pt}  }\asymp      \lim_{h\to\infty}   \Phi(h)      \int_{K g(h)}^{w(h)g(h)}   \Phi_y^h(f(Ay))   e^{ - \int_1^{f(Ay)}  \ov(g(s))ds   }               y^{-1-\alpha}  L(y)     dy
\\
\nonumber
 &\overset{\ref{1fyh}}\geq      \lim_{h\to\infty} \Bigg[   \Phi(h)      \int_{K g(h)}^{w(h)g(h)}   f(y)  e^{-   \int_1^{f(Ay)}   \ov(g(s))ds   }    y^{-1-\alpha}  L(y) dy
\\
\nonumber
  &\hspace{1.75cm}    -     h \Phi(h)      \int_{K g(h)}^{w(h)g(h)}     e^{-   \int_1^{f(Ay)}   \ov(g(s))ds   }    y^{-1-\alpha}  L(y) dy   \Bigg]
\\
\nonumber
   &\overset{ \hspace{2pt}  \phantom{\ref{1fyh}}}{=:}  \hspace{-1pt}   \lim_{h\to\infty} \l[ I_1 - I_2 \r].
  \end{align}
  First we consider $I_2$.  Note that $AK>1$, so  $f(Ay) \geq f(AKg(h)) \geq h$ for all $ y\geq    K g(h)$. Then since $\Phi(h) \asymp  e^{-   \int_1^{h}   \ov(g(s))ds   } $  by  $\l(\ref{phiasymp}\r)$,

  \b{align*}
   \lim_{h\to\infty}    \l| I_2  \r|    &\overset{\phantom{\l(\ref{phiasymp}\r)}}\leq  \hspace{-2pt}   \lim_{h\to\infty} \hspace{-2pt}  h \Phi(h)  \hspace{-3pt}     \int_{K g(h)}^{w(h)g(h)}  \hspace{-7pt}   e^{-   \int_1^{h}   \ov(g(s))ds   }    y^{-1-\alpha}  L(y) dy \hspace{-2pt}

  \lesssim   \hspace{-2pt}    \lim_{h\to\infty}\hspace{-2pt}  h  \hspace{-4pt}   \int_{K g(h)}^{w(h)g(h)}  \hspace{-7pt}       y^{-1-\alpha}  L(y) dy.

  \intertext{By Lemma \mbox{\ref{bigjump}},    $   y^{-1-\alpha} L(y)dy     \asymp    \Pi(dy)$, so as $\ov$ is \mbox{\hyperref[rv]{regularly varying}} at $\infty$,~by~$\l(\ref{condt}\r)$,  }

     \lim_{h\to\infty}    \l| I_2  \r|   &\overset{\ref{bigjump}}\lesssim        \lim_{h\to\infty}  h     \int_{K g(h)}^{w(h)g(h)}    \Pi(dy)

    \overset{\phantom{\ref{bigjump}}}\leq     \lim_{h\to\infty}  h   \ov(Kg(h))    \lesssim    \lim_{h\to\infty}  h \ov(g(h)) \overset{\l(\ref{condt}\r)}=0,
  \end{align*}
    so $I_2=0$, and thus $   \lim_{h\to\infty}  I_1 \leq 0$. As $I_1$ is   non-negative,    $ \lim_{h\to\infty}  I_1 = 0$. Now, changing variables  to $v:=Ay$, as $f$ is \mbox{\hyperref[orv]{$\cal{O}$-regularly varying}}  at $\infty$ and $L$     is \mbox{\hyperref[sv]{slowly varying}} at $\infty$,    by the uniform convergence theorem~\cite[Theorem~1.2.1]{bgt89},
\b{align*}
0 =  \lim_{h\to\infty}      I_1

  &=     \lim_{h\to\infty}   \Phi(h)      \int_{AK g(h)}^{Aw(h)g(h)}   f\l(\f{v}{A}\r)  e^{-   \int_1^{f(v)}   \ov(g(s))ds   }    v^{-1-\alpha} A^{\alpha}  L\l(\f{v}{A}\r) dv
  \\
  &\asymp    \lim_{h\to\infty}   \Phi(h)      \int_{AK g(h)}^{Aw(h)g(h)}   f(v)  e^{-   \int_1^{f(v)}   \ov(g(s))ds   }    v^{-1-\alpha}  L(v) dv.

\intertext{Recall    $v^{-\alpha} L(v) = \ov(v)$, and     $f(v) \asymp   v f'(v)$ uniformly for all  large enough $v$, because   $f$, $f'$ are  $\cal{O}$-regularly varying at $\infty$, see   \cite[Prop 2.10.3]{bgt89}. Then}

   0  &=        \lim_{h\to\infty}   \Phi(h)      \int_{AK g(h)}^{Aw(h)g(h)}   f(v)    e^{-   \int_1^{f(v)}   \ov(g(s))ds   }      v^{-1}  \ov(v) dv
 \\
 &\asymp      \lim_{h\to\infty}   \Phi(h)      \int_{AK g(h)}^{Aw(h)g(h)}     f'(v) \ov(v)    e^{-   \int_1^{f(v)}   \ov(g(s))ds   }     dv.

  \intertext{Now,   one can verify  that $P(v):= \int_{g(1)}^v \ov(u) f'(u)du =  \int_1^{f(v)}  \ov(g(s))ds$ by changing variables from $u$ to $s  =  f(u)$. Then as   $A  >  3$ and $P'(v)  =    \ov(v) f'(v)  \geq~\hspace{-5pt}0$, }

 0  &=          \lim_{h\to\infty}   \Phi(h)     \int_{  AK g(h)}^{A w(h)g(h)}   P'(v)  e^{-  P(v)  }   dv
  \\
       &\geq          \lim_{h\to\infty}    \l[   \Phi(h)  \hspace{-3pt}   \int_{  g(h)}^{w(h)g(h)}    \hspace{-8pt}  P'(v)  e^{-  P(v)  }   dv     -    \Phi(h)    \hspace{-3pt}   \int_{g(h)}^{AK g(h)}    \hspace{-8pt}  P'(v)  e^{-  P(v)  }   dv   \r]

 \\
 &=:  \lim_{h\to\infty} \l[K_1 -K_2  \r] .
  \end{align*}
  Now, recall that by $\l(\ref{j1}\r)$,
  \b{equation} \label{Kexp}
    \lim_{h\to\infty}   K_1 \asymp \lim_{h\to\infty} J_1 \asymp \lim_{h\to\infty}   \l( 1- e^{  -      \int_h^{f(w(h)g(h))}   \ov(g(s))ds}   \r) .
    \e{equation}
So if we  prove $\lim_{h\to\infty} K_1 =0$, then    $ \lim_{h\to\infty}   \int_h^{f(w(h)g(h))}   \ov(g(s))ds =0$,
 and the proof is complete.
  As $K_1$ is always non-negative,  it suffices to   prove that $\lim_{h\to\infty}     K_1    \leq       0$.
  To prove this, we will show  that   $\lim_{h\to\infty}   | K_2|        =     0 $.
  Since $g=f^{-1}$, note   $ f(v) > h$ for $v>g(h)$, then as $\Phi(h) \asymp e^{\int_1^h \ov(g(s))ds}$ by $\l(\ref{phiasymp}\r)$,
  \b{align*}
 \lim_{h\to\infty}   |K_2|    &\overset{\phantom{\l(\ref{phiasymp}\r)}}=     \lim_{h\to\infty}   \Phi(h)  \int_{g(h)}^{AK g(h)}    \ov(v) f'(v)   e^{-   \int_1^{f(v)}   \ov(g(s))ds   }      dv
\\
  &\overset{\l(\ref{phiasymp}\r)}\asymp     \lim_{h\to\infty}        \int_{g(h)}^{AK g(h)}    \hspace{-5pt}    \ov(v) f'(v)  e^{-   \int_h^{f(v)}   \ov(g(s))ds   }     dv

    \leq        \lim_{h\to\infty}           \int_{g(h)}^{AK g(h)}   \hspace{-5pt} v   f'(v)  \ov(v)   v^{-1}     dv.

   \intertext{Recall that by assumption, $v f'(v) \ov(v)$ decreases to 0 as $v\to\infty$, and hence}

    \lim_{h\to\infty}   |K_2|   &\overset{\phantom{\l(\ref{phiasymp}\r)}}\lesssim  \lim_{h\to\infty}  g(h)  \  f'(g(h))  \    \ov(g(h))      \int_{g(h)}^{AKg(h)}   v^{-1}dv
   \\
     &\overset{\phantom{\l(\ref{phiasymp}\r)}}= \lim_{h\to\infty}      g(h)  \  f'(g(h))  \    \ov(g(h))    \times \log(AK) =0.
  \end{align*}

    \end{proof}

\subsubsection{Proof of Corollary \mbox{\ref{entrrepstable}}}

     \b{proof}[Proof of Corollary \mbox{\ref{entrrepstable}}]
    We need to verify that   a stable subordinator of index $\alpha\in(0,1)$ satisfies  $\l(\ref{l6cond}\r)$,   so Theorem \mbox{\ref{entrrep}} applies.    For $t>0$, $x>g(t)+x_0$, by the scaling property of stable subordinators (see \cite[p227]{b98}),
 \b{equation} \label{corollproofbit}
  f_t(x) =    t^{-\f{1}{\alpha}}   f_1\l( \f{x}{t^{\f{1}{\alpha}}}   \r).
 \e{equation}

     \n Now consider the result (see  \cite[Theorem 1.12]{n18})  that for a stable subordinator of index $\alpha\in(0,1)$, $f_1(v) \sim c_\alpha  v^{-1-\alpha}$ as $v\to\infty$, for  $c_\alpha>0$ constant. In~particular, for all large enough $v$, $f_1(v)$ is arbitrarily close to $c_\alpha v^{-1-\alpha}$. Taking e.g.\ $a=2c_\alpha$, it follows that there exist    $a,C \in  (0,\infty)$   such that for all $v>C$, $    f_1(v) \leq a v^{-1-\alpha}$.

       \n  As $\Pi(dv)=u(v)dv = c v^{-1-\alpha}dv$ for a constant $c>0$,   if we can   show that  $x/ t^{1/\alpha}  \geq C$  for all $t>0$, $x>g(t)+x_0$, for a suitable choice of  $x_0>0$,~then~by~$\l(\ref{corollproofbit}\r)$,
      \[
       f_t(x)   =  t^{-\f{1}{\alpha}}   f_1\l( \f{x}{t^{\f{1}{\alpha}}}   \r)    \leq  ac t  x^{-1-\alpha}  = A t u(x),
      \]
  for $A=ac$,   so   condition $\l(\ref{l6cond}\r)$ is satisfied, and the proof will be complete.

 Indeed, by $\l(\ref{condt}\r)$, $\lim_{t\to\infty} t\ov(g(t))= \lim_{t\to\infty} tg(t)^{-\alpha}  =0$,     so there exists  $D\in(0,\infty)$    such that  for all $t>D$,        $tg(t)^{-\alpha}   \leq   C^{-\alpha}$,   so    $t^{1/\alpha} \leq  C^{-1} g(t)$, and hence for all $t>D$,
\[
\f{x}{t^{\f{1}{\alpha}} }   \geq   \f{g(t) + x_0}{t^{\f{1}{\alpha}}}  \geq   \f{g(t)}{t^{\f{1}{\alpha}}}  \geq  C   .
 \]

 \n On the other hand, if $t\leq D$, then
 $  x/t^{1/\alpha} \geq  (g(t)+  x_0 ) /  D^{1/\alpha}   \geq   x_0  /  D^{1/\alpha}  $, and
 (choosing $x_0$ large enough that
$x_0 / D^{1/\alpha} > C$ if necessary),
 we conclude that  $x/t^{1/\alpha}>C$   for all      $t>0$, $ x> g(t)+x_0$, so    \cite[Theorem 1.12]{n18}  applies to $\l(\ref{corollproofbit}\r)$. It follows that   condition   $\l(\ref{l6cond}\r)$ is satisfied, and so  Theorem \mbox{\ref{entrrep}} applies,~as~required.

     \e{proof}

\section{Proof of Lemma  \hbox{\ref{lemma1}}} \label{lemma1proof}
  Before proving Lemma \mbox{\ref{lemma1}}, let us begin by stating
  Lemma  \hbox{\ref{keylemma}}, which is key to proving Lemmas \mbox{\ref{lemma1}} and \mbox{\ref{lemma2}}, and is itself proven in Section~\hbox{\ref{lemmasproofs}}.

   \b{lemma} \label{keylemma}
   Let $(X_t)_{t\geq0}$ be a subordinator satisfying the assumptions in case (\hbox{\hyperref[case1]{i}}) or (\hbox{\hyperref[case2]{ii}}). Then there exists a constant $C>0$, which depends only on the law of $X$, such that for all $A(t)\in(1,\infty)$,    $B(t)>0$,  and  $ H(t) \in (0,1) $,
      \b{equation}   \label{keyeqn}
   \pp\l( X_t^{ (0,A(t))}  \hspace{-2pt} > \hspace{-2pt} B(t) \r) \leq  \exp \hspace{-1pt} \l(   \hspace{-1pt} C  t \log\l( \hspace{-0.5pt} \f{1}{H(t)} \hspace{-0.5pt} \r)   H(t)^{-\f{A(t)}{B(t)}}    \ov(A(t)) \f{A(t)}{B(t)}   \r) \hspace{-0.5pt}  H(t).
   \e{equation}
                 \e{lemma}

  \b{proof}[Proof of Lemma  \hbox{\ref{lemma1}}]          Lemma
\mbox{\ref{lemma1}} is simpler to prove in
case (\hbox{\hyperref[case2]{ii}}) than   case (\hbox{\hyperref[case1]{i}}) thanks to the     condition $\l(\ref{cond2}\r)$. We  thus omit the   proof  in~case~(\hbox{\hyperref[case2]{ii}}).

      Since $g      =     f^{-1}$ is continuous, for each $x  \geq  0$,~$t  >    0$,
  \b{equation}
  \label{s-}
  \pp\big(\cal{O}_{t, X^{(0,x)}}\big)     =      \pp\big(\cal{O}_{t-, X^{(0,x)}}\big)   ,
  \end{equation}  where  $\cal{O}_{t-, X^{(0,x)}} := \bigcap_{u<t} \cal{O}_{u, X^{(0,x)}} $, and moreover by the  definition $\l(\ref{truncoo}\r)$,
    \b{equation}
  \label{subevent}
  \pp(\cal{O}_{t, X^{(0,x)}}  )     \leq   \pp\big( X_t^{(0,x)} > g(t) \big).
  \e{equation}

  Now, we partition and disintegrate on the value of $\Delta_1^{g(t)}$, which is exponentially distributed with rate $\ov(g(t))$,     then note   by $\l(\ref{o}\r)$, $\l(\ref{s-}\r)$ and $\l(\ref{truncoo}\r)$, that $  \pp( \oo_{t} |  \Delta_1^{g(t)}   =  s  )   =    \pp( \oo_{s} |  \Delta_1^{g(t)}   =  s  )   =     \pp( \oo_{s-} |  \Delta_1^{g(t)}   =  s  )   =   \pp(\oo_{s,X^{(0,g(t))}}    )  $, so
  \begin{align}
  \nonumber
  \pp\l( \oo_{t}  \r) &=   \pp( \oo_{t} ;  \Delta_1^{g(t)} \leq t  )  +   \pp( \oo_{t}  ;  \Delta_1^{g(t)} >t   )

 \\

  \nonumber
  &=  \ov(g(t))    \int_0^t      \pp( \oo_{t} |  \Delta_1^{g(t)} =s  )   e^{ -  \ov(g(t)) s} ds +  \pp( \oo_{t}  ;  \Delta_1^{g(t)} >t   )

 \\

  \label{oneone}
  &=  \ov(g(t))    \int_0^t   \pp ( \   \oo_{s,X^{(0,g(t))}}    \    )   \  e^{ -  \ov(g(t)) s}   ds   +   \pp( \oo_{t}  ;  \Delta_1^{g(t)} >t   )  .

\intertext{Now, observe    that by the definition   $\l(\ref{truncoo}\r)$, $  \pp( \oo_{s} |  \Delta_1^{g(t)} >s  ) = \pp(\oo_{s,X^{(0,g(t))}}    )  $, so}

  \nonumber
\pp(\oo_s )
&= \pp(   \oo_s ; \Delta_1^{g(t)} \leq s ) +  \pp(   \oo_s ; \Delta_1^{g(t)} > s )

\\
  \nonumber

&=  \pp(   \oo_s ; \Delta_1^{g(t)} \leq s ) +  \pp(   \oo_s | \Delta_1^{g(t)} > s ) \pp(\Delta_1^{g(t)} > s )

\\
  \label{twotwo}

&= \pp(   \oo_s ; \Delta_1^{g(t)} \leq s ) + \pp\l(\oo_{s,X^{(0,g(t))}}    \r)   e^{ -  \ov(g(t)) s}.
\intertext{Disintegrating  on  $\Delta_1^{g(t)} $,   recalling the notation $\l(\ref{Phi}\r)$, $\l(\ref{D}\r)$, by  $\l(\ref{oneone}\r)$, $\l(\ref{twotwo}\r)$,~and~$\l(\ref{s-}\r)$,}

\nonumber
  \pp\l( \oo_{t}  \r)   &\overset{\phantom{\l(\ref{s-}\r)}}= \ov(g(t)) \int_0^t    \big[   \mathbb{P}   (\cal{O}_{s}) - \pp(   \oo_s ; \Delta_1^{g(t)} \leq s ) \big] ds    +  \pp( \oo_{t}  ;  \Delta_1^{g(t)} >t   )

\\

  \nonumber
    &\overset{\l(\ref{s-}\r)}= \ov(g(t))   \Phi(t)
- \ov(g(t))^2   \int_0^t      \int_0^s       \mathbb{P}   (\cal{O}_{v,X^{(0,  g(t))}  })   e^{-\ov(g(t))v}   dv
ds      +  \pp( \oo_{t}  ;  \Delta_1^{g(t)} >t   )

\\
  \label{123}
& \overset{\phantom{\l(\ref{s-}\r)}}= \hspace{-3pt}: (a1) - (a2) + (a3).

\intertext{One easily     verifies that $(a2) = o(\ov(g(t))\Phi(t))$ as $t\to\infty$, using     $\l(\ref{condt}\r)$.  For   $t\geq1$,}
\label{Phiconstant}
      \Phi(t)  &\geq   \Phi(1) = \textup{constant} >0.
      \intertext{We  use $\l(\ref{Phiconstant}\r)$ to prove   $ (a3) = o(\Phi(t)\ov(g(t)))$ as $t\to\infty$. We partition $(a3)$  as}

\nonumber
 (\mbox{\hyperref[123]{$a$3}}) \ \ &\hspace{-0.2cm}= \hspace{-0.05cm} \mathbb{P}(\oo_{t} ; \D^{\theta g(t)} \hspace{-0.1cm}<  \hspace{-0.05cm} t; \D^{  g(t)}   \hspace{-0.1cm}  >   \hspace{-0.1cm}  t)  \hspace{-0.05cm}      +   \hspace{-0.05cm}   \mathbb{P}(\oo_{t} ; \D^{g_1(t)}  \hspace{-0.05cm}  <  \hspace{-0.05cm}   t ;\D^{\theta g(t)}>t )     \hspace{-0.05cm}   +   \hspace{-0.05cm}    \mathbb{P}(\oo_{t} ; \D^{g_1(t)} \hspace{-0.1cm} >  \hspace{-0.05cm}  t)

\\

\label{3ABC}
     \ \ &\hspace{-0.2cm}=:   (3A)   +  (3B)  +   (3C),

\intertext{with $g_1(t):= g(t)/\log(t)$, and $\theta \in (0,1)$. Later we   split $(3B) $ into more~pieces. }

\intertext{\textbf{Proof for ($\mbox{\hyperref[3ABC]{$3A$}}$)}  Disintegrating on   $  \D^{\theta g(t)}$, which is exponentially distributed,}

   \nonumber
   (\mbox{\hyperref[3ABC]{$3A$}}) &= e^{-t\ov(g(t))} \mathbb{P}(\oo_{t} ; \D^{\theta g(t)}<t | \D^{  g(t)}>t)

\\

   \nonumber
&=\hspace{-0.05cm}  e^{-t\ov(g(t))}   \hspace{-0.175cm} \int_0^t  \hspace{-0.125cm}    \mathbb{P}(\oo_{t} | \D^{\theta g(t)} \hspace{-0.05cm} =   \hspace{-0.05cm}  s; \D^{  g(t)}\hspace{-0.05cm}  >   \hspace{-0.05cm}  t)    \Pi((\theta g(t)  \hspace{-0.025cm} ,g(t) ) ) e^{\hspace{-0.025cm} -   ( \ov(\theta g(t)) - \ov(g(t) ) ) s } ds

\\

   \nonumber
&\leq      ( \ov(\theta g(t)) - \ov(g(t) ) ) \int_0^t      \mathbb{P}(\oo_{t} | \D^{\theta g(t)}  = s; \D^{  g(t)}>t)  ds

\\

&\leq  ( \ov(\theta g(t))    -    \ov(g(t) ) )         \int_0^t      \mathbb{P}(\oo_{s,X^{(0,\theta g(t))}  })  ds

\label{orvl}

 \leq      \l( \f{\ov(\theta g(t)) }{ \ov(g(t) ) }     -   1 \r)  \ov(g(t))        \int_0^t      \mathbb{P}(\oo_{s   })  ds  .
\end{align}

\n Now,   $\lim_{\theta\to1} \lim_{t\to\infty} \l(   \ov(\theta g(t))/\ov(g(t))     -1\r)=0$, as  $\ov$ is \mbox{\hyperref[crv]{CRV}} at $\infty$, and therefore  by $\l(\ref{orvl}\r)$,    $(\mbox{\hyperref[3ABC]{$3A$}})    =     o(\ov(g(t))\Phi(t))$ as $t\to\infty$ then   $ \theta \to1$.

  \p{Partitioning (\mbox{\hyperref[3ABC]{$3B$}})} Recall   notation $\l(\ref{o}\r)$. Disintegrating on $\D^{g_1(t)}  , $   by $\l(\ref{subevent}\r)$,
  \b{align}
\nonumber
(\mbox{\hyperref[3ABC]{$3B$}}) &\overset{\phantom{\l(\ref{subevent}\r)}}= \mathbb{P}(\oo_{t} ; \D^{g_1(t)}<t ;\D^{\theta g(t)}>t )

\\

\nonumber
 &\overset{\phantom{\l(\ref{subevent}\r)}}=    \ov(g_1(t)) \int_0^t  \pp(\oo_{t} ; \D^{\theta g(t)}>t    | \D^{g_1(t)}  = s) e^{ - \ov( g_1(t) )  s    } ds

\\

\nonumber
 &\overset{\l(\ref{subevent}\r)}\leq  \ov(g_1(t)) \int_0^t  \pp(\oo_{s } ;     X_t > g(t) ; \D^{\theta g(t)}>t    | \D^{g_1(t)}  = s)   ds

\\

\nonumber
 &\overset{\phantom{\l(\ref{subevent}\r)}}=   \hspace{-0.15cm}  \ov(g_1(t))  \hspace{-0.15cm}  \int_0^t    \hspace{-0.175cm}   \pp(\oo_{s } ;    \hspace{-0.05cm}    \hat{X}_{t-s}^{(0,\theta g(t))}   \hspace{-0.1cm}   +   \hspace{-0.05cm} X_{s-}   \hspace{-0.1cm} +  \hspace{-0.05cm}  (   \hspace{-0.05cm}  X_s   \hspace{-0.05cm} -  \hspace{-0.05cm}  X_{s-}   \hspace{-0.05cm}  )    \hspace{-0.05cm} >  \hspace{-0.05cm}  g(t) ; \D^{\theta g(t)}  \hspace{-0.1cm} >  \hspace{-0.05cm} t    | \D^{g_1(t)}    \hspace{-0.1cm} =   \hspace{-0.1cm}  s    \hspace{-0.025cm}   )  ds,

\intertext{where $\hat{X}^{(0,\theta g(t))}$ is an independent copy of $X$ with truncated L\'evy measure as in the definition $\l(\ref{truncoo}\r)$. Then as the jump at time  $\D^{g_1(t)} $ has size   $  X_s - X_{s-} \leq    \theta g(t)$,  }

 \nonumber
(\mbox{\hyperref[3ABC]{$3B$}}) &  \hspace{-0.05cm}   \leq   \hspace{-0.05cm}    \ov(g_1(t))   \hspace{-0.1cm}  \int_0^t   \hspace{-0.15cm}  \pp(\oo_{s,X^{(0,g_1(t))} } ;     \hat{X}_{t-s}^{(0,\theta g(t))}  \hspace{-0.05cm}  +  \hspace{-0.05cm}  X_{s-}^{(0,g_1(t))}   \hspace{-0.05cm} >  \hspace{-0.05cm}  (1  \hspace{-0.05cm} -  \hspace{-0.05cm} \theta)g(t) ; \D^{\theta g(t)}  \hspace{-0.05cm} >  \hspace{-0.05cm} t    )  ds

\\

\nonumber
 &\leq  \ov(g_1(t)) \int_0^t  \pp(\oo_{s,X^{(0,g_1(t))} } ;     \hat{X}_{t-s}^{(0,\theta g(t))} + X_{s-}^{(0,g_1(t))} > (1-\theta)g(t)    )  ds.

\intertext{Partitioning according to  $A:= \{ X_s^{(0,g_1(t))} > (1-\theta) g(t) /2 \} $ and $A^c$,}

\nonumber
 (\mbox{\hyperref[3ABC]{$3B$}}) &\leq\ov(g_1(t)) \int_0^t  \pp(\oo_{s,X^{(0,g_1(t))} } ;     \hat{X}_{t-s}^{(0,\theta g(t))} + X_{s-}^{(0,g_1(t))} > (1-\theta)g(t)  ;A  )  ds \quad

\\

\nonumber
 &+\ov(g_1(t)) \int_0^t  \pp(\oo_{s,X^{(0,g_1(t))} } ;     \hat{X}_{t-s}^{(0,\theta g(t))} + X_{s-}^{(0,g_1(t))} > (1-\theta)g(t) ;A^c    )  ds \quad

\\

\nonumber
 &\leq  \ov(g_1(t))    \int_0^t   \pp\l(  X_s^{(0,g_1(t))} >   \f{ (1-\theta) g(t) }{2}  \r) ds \quad

\\

\nonumber
  &+ \ov(g_1(t)) \int_0^t  \pp\l(  \oo_{s,X^{(0,g_1(t))}}  ;  \hat{X}_{t-s}^{(0,\theta g(t))}  > \f{(1-\theta)g(t) }{2}    \r)  ds \quad

\\

\nonumber
&\leq  t \ov(g_1(t))    \pp\l(  X_t^{(0,g_1(t))} >   \f{ (1-\theta) g(t) }{2}  \r)

\\

\nonumber
 &+  \ov(g_1(t))  \mathbb{P}\l( X_{t}^{(0,\theta g(t))}  > \f{(1-\theta)g(t) }{2}    \r)    \int_0^t  \pp\l(  \oo_{s }  \r)  ds.

\\

\label{j1j2}
   &=:(J1) + (J2).
   \end{align}

\p{Proof for (\mbox{\hyperref[j1j2]{$J1$}})}

By Potter's theorem \cite[Theorem 1.5.6]{bgt89},       $\ov(g_1(t)) \lesssim   \log(t)^{2\alpha} \ov(g(t)) $ as $t\to\infty$. Applying Lemma~\hbox{\ref{keylemma}} with $H(t)      =     t^{-n},n    >     1$,~as~$t\to\infty$,

\begin{align*}
(\mbox{\hyperref[j1j2]{$J1$}})    &\overset{\phantom{\ref{keylemma}}}\lesssim  t \log(t)^{2\alpha}    \ov(g(t))    \pp\l(  X_t^{(0,g_1(t))} >   \f{ (1-\theta) g(t) }{2}  \r)

   \overset{\ref{keylemma}}\lesssim    t \log(t)^{2\alpha}    \ov(g(t))    \exp\l(    (*)     \r)  t^{-n},

\\
   (*)      &\overset{\phantom{\ref{keylemma}}}\lesssim  \hspace{-0.05cm}    \f{  2n t \log(t)   }{   (1-\theta) g(t)   } t^{\f{2n}{(1-\theta )\log(t)}}       \ov(g_1(t))   \f{2 g_1(t)  }{ (1-\theta) g(t) }    \hspace{-0.05cm}    \lesssim  \hspace{-0.05cm}     t   e^{\f{2n}{1-\theta}}      \ov(g_1(t))

 \hspace{-0.05cm}  \lesssim \hspace{-0.05cm}    t \ov(g_1(t))   .
\end{align*}
By   $\l( \ref{condt}  \r)$, $ \lim_{t\to\infty}(*)=0$ for each   $n>1$ and $\theta\in(0,1)$, so  in case (\hbox{\hyperref[case1]{i}}),     by $\l(\ref{Phiconstant}\r)$,
\(
(\mbox{\hyperref[j1j2]{$J1$}}) \lesssim   \log(t)^{\alpha+\d}    \ov(g(t))      t^{-(n-1)}    = o(\ov(g(t)\Phi(t))
\)
as $t\to\infty$ then   $ \theta \to 1$.

\begin{comment}
\n In   case (\hbox{\hyperref[case2]{ii}}), let us replace $g_1(t)$ by  $cg(t)$, for  $c\in(0,\theta)$, and $(\mbox{\hyperref[j1j2]{$J1$}})$ by  $(J1')$  with $cg(t)$ in place of $g_1(t)$.
As  $\ov$ is \mbox{\hyperref[orv]{$\cal{O}$-regularly varying}} at $\infty$,    as $t\to\infty$,
\b{align*}
(J1') &=   t \ov(c g(t))    \pp\l(  X_t^{(0,cg(t))   } > \f{(1-\theta) g(t) }{ 2  }    \r)
\\
   &\lesssim  t \ov(g(t))\pp\l(  X_t^{(0,cg(t))   } > \f{(1-\theta) g(t) }{ 2  }    \r).

\intertext{By   Lemma \hbox{\ref{keylemma}} with $H(t)=t^{-n}$, for $n$, $\eps$ such that $n>1+\eps>1$, as $t\to\infty$,}

(J1')   &\lesssim        \ov(g(t))   \exp\l(     (*)    \r)   t^{-(n-1)},

\\
(*) &\lesssim t    \log(t^{n})    t^{\f{2n}{(1-\theta)\log(t)}}         \ov(cg(t))      \f{2 c g(t)  }{  (1-\theta) g(t)}

\\

&= t    n  \log(t)    e^{\f{2n}{1-\theta}}        \ov(cg(t))

\\

&\lesssim     t \log(t)         \ov(g(t))

\\

 &\lesssim      t^{1+\eps}        \ov(g(t)).
\end{align*}
By $\l(\ref{cond2}\r)$, $\lim_{t\to\infty}(*)=0$ by       so for each fixed $\theta\in(0,1)$, $(J1') = o(\ov(g(t)))$ as $t  \to  \infty$ in~case~(\hbox{\hyperref[case2]{ii}}), and by $\l(\ref{Phiconstant}\r)$,   $(J1')   =   o(\ov(g(t))\Phi(t))$ as $t\to\infty$ then $ \theta   \to  1$.
\end{comment}

\p{Proof for (\mbox{\hyperref[j1j2]{$J2$}})}
Applying Markov's inequality,
\b{align*}
(\mbox{\hyperref[j1j2]{$J2$}}) &=    \ \  \   \ov(g_1(t))  \Phi(t)
 \mathbb{P}\l( X_{t}^{(0,\theta g(t))}  > \f{(1-\theta)g(t) }{2}    \r)

\leq    \    \f{   2   \ov\l(    g_1(t)\r)    \Phi(t)   }{   (1-\theta) g(t)}
\mathbb{E}\l[   X_t^{(0,\theta g(t))}  \r]

\\

&=      \     \f{   2  t   \ov\l(    g_1(t)\r)   \Phi(t)   }{   (1-\theta) g(t)}
\int_0^{\theta g(t)} x \Pi(dx)

=    \     \f{   2  t   \ov\l(    g_1(t)\r)   \Phi(t)   }{   (1-\theta) g(t)}
\int_0^{\theta g(t)} \l(   \ov(x) - \ov(\theta g(t))\r) dx

\\

&
\leq     \     \f{   2 t \ov(g_1(t))   \Phi(t)     }{   (1-\theta) g(t)}
\int_0^{\theta g(t)}   \ov(x)dx.

\intertext{As $\hspace{-1pt}\ov\hspace{-1pt}$ is \mbox{\hyperref[rv]{regularly varying}} at $\hspace{-1pt}\infty\hspace{-1pt}$, \hspace{-5.5pt} by Karamata\hspace{-1pt}'\hspace{-1pt}s theorem \hspace{-6pt} \cite[\hspace{-2pt}Prop1.5.8]{bgt89}~\hspace{-5pt}~as~\hspace{-3.5pt}~$ t\hspace{-2.5pt}\to\hspace{-3pt}\infty,$}

(\mbox{\hyperref[j1j2]{$J2$}}) &\lesssim           \f{   2   t \ov(g_1(t)) \Phi(t)  }{   (1-\theta) g(t)}    \theta g(t)   \ov(\theta g(t))   \lesssim    t\ov(g_1(t))    \Phi(t) \ov(g(t)).

\intertext{By $\l(\ref{condt}\r)$,  $\lim_{t\to\infty}    t \ov(g_1(t))   =0$,  so $ (\mbox{\hyperref[j1j2]{$J2$}})=o(\ov(g(t))\Phi(t))$ as $t\to\infty$ for each fixed $\theta\in(0,1)$, and    $(\mbox{\hyperref[j1j2]{$J2$}}) = o(\ov(g(t))\Phi(t))$ as $t\to\infty$ then $  \theta \to1$.}

\end{align*}
\begin{comment}
\intertext{For  case (\hbox{\hyperref[case2]{ii}}), recall   we have $(J2')$ with $c g(t)$ in place of $g_1(t)$, for $c\in (0,\theta)$. By  Lemma \mbox{\ref{keylemma}} with $H(t)= 1/\log(\log(t))$, since $\ov(c(g(t))\lesssim \ov(g(t))$ as $t\to\infty$,}

  (J2')     &=    \ov(cg(t))  \Phi(t)     \mathbb{P}\l( X_{t}^{(0,\theta g(t))}  > \f{(1-\theta)g(t) }{2}    \r)

  \\
  &\lesssim  \f{\ov(  g(t) ) \Phi(t)}{\log(\log(t))} \exp\l( (*)        \r)  ,

 \\

 (*)   &=   t    \log(\log(\log(t)  )  )      \log(\log(t))^{\f{  2\theta   }{1-\theta}}     \ov( \theta g(t) )          \f{2\theta }{ 1-\theta}

    \lesssim \hspace{-0.05cm}    t \log(t)^{\f{\alpha}{2}    }    \ov(g(t) )      .
    \end{comment}

 \p{Proof for    (\mbox{\hyperref[3ABC]{$3C$}})}
 By $\l(\ref{subevent}\r)$ and Lemma \hbox{\ref{keylemma}} with~$H(t)\hspace{-2pt}  = \hspace{-2pt}  \ov(g(t))^{2}$,~as~$t \hspace{-2.5pt}~\to~\hspace{-2.5pt}\infty$,
\b{equation} \label{3Cref}
(\mbox{\hyperref[3ABC]{$3C$}}) = \mathbb{P}(\oo_{t} ; \D^{g_1(t)}>t) \leq \pp\l(   X_t^{(0,g_1(t))}   > g(t) \r)
\leq  \exp\l(  (*)   \r)     \ov(g(t))^2  ,
\e{equation}
\b{align*}
   (*) &\lesssim      t    \log\l(\ov(g(t))^{-2}\r)    \ov(g(t))^{    -\f{ 2  }{ \log(t) }  }      \f{ \ov(g_1(t))}{\log(t)}

  \lesssim      \f{t  \log\l(  \f{1}{ \ov(g(t))}   \r)}{ \log(t)  }   \l(  \f{1}{ \ov(g(t))}   \r)^{  \f{1}{\log(t)}   }     \ov(g_1(t)) .

\intertext{Now we    split into subsets of $t>0$ (if one is bounded,   we need only consider the other).

  Fix     $M> 6/\alpha$. First we consider all $t>0$ for which $g(t)\leq t^M$. Note $\ov(x)=x^{-\alpha} L(x)$ with  $L$   \mbox{\hyperref[sv]{slowly varying}} at $\infty$. Then as $g(t)^{1/M}  \leq   t  \to  \infty$,}

(*)  &\lesssim     \f{   \log\l(   g(t)^{\alpha} L(g(t))^{-1}   \r)}{ \log(t)  }    g(t)^{\f{\alpha}{\log(t)}} L(g(t))^{ - \f{1}{\log(t)}   }   .

\intertext{As $L$ is slowly varying at $\infty$, $g(t)^{\alpha} L(g(t))^{-1} \leq  g(t)^{2\alpha}$,   so as $g(t)^{1/M}  \hspace{-2.5pt}  \leq  \hspace{-1pt}   t\to\infty$,}

(*) &\lesssim \f{   \log\l(   g(t)^{2\alpha}   \r)}{ \log(t)  }    g(t)^{\f{2\alpha}{\log(t)}}

  \lesssim \f{   \log\l(  t^{2\alpha M}  \r)}{ \log(t)  }  t^{ \f{ 2\alpha M}{\log(t)}}    = 2\alpha Me^{2\alpha M}     < \infty .

\intertext{Therefore for all $t>0$ for which $g(t)\leq t^M$,  we can conclude by  $\l(\ref{3Cref}\r)$ and $\l(\ref{Phiconstant}\r)$ that  $(\mbox{\hyperref[3ABC]{$3C$}})\lesssim \ov(g(t))^2  = o(\ov(g(t))\Phi(t))$ as $t\to\infty$ then $\theta \to1$.}

\intertext{For all $t>0$ such that $g(t)\geq t^M$,  as $t\to\infty$, $\log(1/\ov(g(t))) \leq \ov(g(t))^{-1/3}$, $\ov(g(t))\leq t^{-\alpha M/2}$, and $\ov(g(t))^{-1/\log(t)}\leq \ov(g(t))^{1/3}$. Applying Potter's      theorem \mbox{\cite[Theorem 1.5.6]{bgt89}} to $\ov(g_1(t))$,   for each $\d>0$,  as $g(t)^{1/M} \geq t\to\infty$,}

(*) &\lesssim   \f{t  \log\l(  \f{1}{ \ov(g(t))}   \r)}{ \log(t)  }   \l(  \f{1}{ \ov(g(t))}   \r)^{  \f{1}{\log(t)}   }     \ov(g_1(t))

\lesssim  \f{t    \ov(g(t))^{-\f{1}{3}}  }{ \log(t)  }  \ov(g(t))  ^{ - \f{1}{3}  }     \ov(g_1(t))
\\
&\lesssim  \f{t    \ov(g(t))^{\f{1}{3}}  }{ \log(t)  }     \log(t)^{-\alpha+\d}

\hspace{84.5pt} \lesssim    t^{1-\f{1}{6}   \alpha  M}     \log(t)^{-\alpha+\d}.

\intertext{Recall   $M>6/\alpha$, so $1- \alpha M /6 <0$, and  $\lim_{t\to\infty}(*)=0$. Then by $\l(\ref{Phiconstant}\r)$ and $\l(\ref{3Cref}\r)$,       $(\mbox{\hyperref[3ABC]{$3C$}})=o(\ov(g(t))\Phi(t))$ as $t\to\infty$ then $ \theta\to1$ in case~(\hbox{\hyperref[case1]{i}}), as required.}
  \end{align*}

\begin{comment}
\intertext{In  case (\hbox{\hyperref[case2]{ii}}), recall   we have $(3C')$ with $c g(t) $ in place of $g_1(t)$. By $\l(\ref{subevent}\r)$, applying  Lemma   \hbox{\ref{keylemma}} with $   H(t) = \ov(g(t))^2   $, for arbitrarily small $\eta>0$, as $t\to\infty$,}

 (3C')&=   P( \cal{O}_t ;   \Delta_1^{c g(t)} > t )   \overset{(\ref{subevent})}\leq
  \pp(X_t^{(0,c g(t))} > g(t) )

  \\

  &\leq \exp\l(   (*)     \r)      \ov(g(t))^2,
\\
 (*)& =       t    \log\l(\ov(g(t))^{-2}\r)    \ov(g(t))^{    - 2c }      c  \ov(c g(t))
 \\
 &\lesssim     t  \log(  1/ \ov(g(t)))            \ov(g(t))^{-2c}      \ov(g(t))

\\

 &\lesssim   t   \ov(g(t))^{1-\eta -2c}        .
  \end{align*}
 By $\l(\ref{cond2}\r)$, taking $c,\eta$ sufficiently small,   $\lim_{t\to\infty}(*)  =  0$,   then by $\l(\ref{Phiconstant}\r)$, $(3C')  \lesssim   \ov(g(t))^2  =   o(\ov(g(t))\Phi(t))$ as $t  \to  \infty$ then $ \theta  \to  1$ in case (\hbox{\hyperref[case2]{ii}}), as required.
\end{comment}

\e{proof}

\section{Proof of Lemma \hbox{\ref{lemma2}}}   \label{lemma2proof}
     In order to prove Lemma \hbox{\ref{lemma2}}, we    require Lemma  \hbox{\ref{1.1}}, which is proven in Section~\hbox{\ref{lemmasproofs}}.

   \b{lemma} \label{1.1} \hspace{-8pt} Recalling $\l(\ref{gyh}\r)$ and $\l(\ref{t0y}\r)$, if $t   >   t_0(y)$,   then $g_{y}^{h}(t)     \geq      \l(    1    -    1/A    \r)g(t)$.

  \e{lemma}

  \n Lemma \hbox{\ref{lemma2}} shall be proven by  splitting up $\rho_y^h(t)$ into   smaller pieces, and then showing that the inequalities $\l(\ref{1}\r)$ and $\l(\ref{2}\r)$ hold for each piece separately.

  \b{proof}[Proof of Lemma  \hbox{\ref{lemma2}}] As with      Lemma
\mbox{\ref{lemma1}},  Lemma \mbox{\ref{lemma2}}
is simpler to prove in
case (\hbox{\hyperref[case2]{ii}}) than   case (\hbox{\hyperref[case1]{i}}) thanks to the     condition $\l(\ref{cond2}\r)$. We  thus omit the   proof  in~case~(\hbox{\hyperref[case2]{ii}}).

  Recall the notation introduced in  $\l(\ref{phiyh}\r)$ and  $\l(\ref{truncoo}\r)$.

 First, repeating the argument  as  in $\l(\ref{123}\r)$, but now with $g_{y}^{h}(t)$ in place of $g(t)$,
  \b{align}    \nonumber
   \pp( \cal{O}_{t}^{g_{y}^{h}}) &= \ov(g_{y}^{h}(t))   \Phi_y^h(t)       +   \pp\l(   \hspace{-2pt} \oo_{t}^{g_{y}^{h}}  ;  \Delta_1^{g_{y}^{h}(t)} >t   \hspace{-2pt} \r)

   - \ov(g_{y}^{h}(t))^2 \int_0^t\int_0^s \mathbb{P}   (\cal{O}_{v}^{g_{y}^{h}})   e^{-\ov(g_{y}^{h}(t))v}   dv
ds
\\  \nonumber
&\leq   \ov(g_{y}^{h}(t))     \Phi_y^h(t)      + \pp\l(\oo_{t,X^{(  0, g_{y}^{h}(t) ) } }^{g_{y}^{h}} \r)

- \ov(g_{y}^{h}(t))^2 \int_0^t\int_0^s \mathbb{P}   (\cal{O}_{v}^{g_{y}^{h}})   e^{-\ov(g_{y}^{h}(t))v}   dv ds
\\       \label{triv}
&\leq   \ov(g_{y}^{h}(t))     \Phi_y^h(t)      + \pp\l(\oo_{t,X^{(  0, g_{y}^{h}(t) ) } }^{g_{y}^{h}} \r)    .
  \end{align}

\begin{comment}
  \n  Now, consider the property that $\pp(\oo_{t}) = \pp( \oo_{t-}   )$, where $\oo_{t-}:= \bigcap_{0\leq s<t}\oo_s$. This is easily deduced from  the fact that $X$ has left-continuous sample paths, almost surely.    Then it follows that
   \b{equation}   \label{-}
    \pp\l(     \oo_{v,X}^{g_{y}^{h}} | \D^{ g_{y}^{h}(t)}=v   \r)  =  \pp\l(     \oo_{v-,X}^{g_{y}^{h}} | \D^{ g_{y}^{h}(t)}=v   \r)    =    \pp\l(     \oo_{v,X^{(0, g_{y}^{h}(t))}}^{g_{y}^{h}}    \r)  .
  \e{equation}

  \end{comment}
Recall the notation $\l(\ref{D}\r)$ and  $\l(\ref{rhoyh}\r)$. By  $\l(\ref{triv}\r)$, partitioning on the value of $\D^{\f{g_{y}^{h}(t)}{\log(t)}   } $,
\b{align}
\nonumber
\rho_y^h(t) : \hspace{-6pt} &\overset{\phantom{\l(\ref{triv}\r)}}{=} \f{ \pp(  \cal{O}_t^{g_{y}^{h}}   )  }{  \Phi_y^h(t)   } -   \ov(g_{y}^{h}(t))

   \overset{\l(\ref{triv}\r)}\leq    \f{1}{\Phi_y^h(t)}   \pp\l(\oo_{t,X^{(  0, g_{y}^{h}(t) ) }  }^{g_{y}^{h}} \r)
    \\
    \nonumber
      &\overset{\phantom{\l(\ref{triv}\r)}}=     \f{1}{\Phi_y^h(t)}  \Big[   \pp     \Big(      \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ; \D^{\f{g_{y}^{h}(t)}{\log(t)}   }      > t    \Big) \hspace{-2pt}  +  \hspace{-2pt}  \pp   \Big(   \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ; \D^{\f{g_{y}^{h}(t)}{\log(t)}   }        \leq        t   \Big) \Big]
\\
\label{ab}
     &\overset{\phantom{\l(\ref{triv}\r)}}{=} \hspace{-3pt}:    \f{1}{\Phi_y^h(t)}  \l[(a)+(b) \r].
     \end{align}

\n So   to prove  $\l(\ref{1}\r)$, we  need to prove, uniformly in $h>0,y>g(h)$, as $t\to\infty$,
\b{align}
\label{1'}
(a)+(b) &\lesssim   \f{\Phi_y^h(t)}{t\log(t)^{1+\eps}} \l( 1 + \f{1}{f(y)-h} \r) .

  \intertext{For $\l(\ref{2}\r)$, we  need        suitable $u$ so that uniformly in $h>0,y>g(h)$, as $t\to\infty$,}

\label{2'}
(a)+(b) &\leq  \Phi_y^h(t) u(t) \ov(g(t))  \l( 1 + \f{1}{f(y)-h} \r) .
\end{align}

   \p{Proof for (\mbox{\hyperref[ab]{$a$}})
   }
   Recall the notation $\l(\ref{ogyh}\r)$,  $\l(\ref{truncoo}\r)$. By Lemma \mbox{\ref{1.1}},  for all $h>0$, $y>g(h)$,  $g_y^h(t)/\log(t) \geq (1-A^{-1})g(t)/\log(t)> 1$  as $t\to\infty$,    so by $\l(\ref{subevent}\r)$, Lemma \mbox{\ref{1fyh}},  and Lemma \hbox{\ref{keylemma}}   with $H(t) =  t^{-n}  $, for   $n>1$,  uniformly in $h>0$, $y>g(h)$,   as~$t\to\infty$,

   \b{align}
   \nonumber
   (\mbox{\hyperref[ab]{$a$}})     &\overset{\phantom{\ref{1fyh}}}= \pp     \Big(      \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ; \D^{\f{g_{y}^{h}(t)}{\log(t)}   }      > t    \Big)

    \hspace{17pt}   \overset{\l(\ref{subevent}\r)}\leq      \pp\Big(   X_t^{(0,\f{g_{y}^{h}(t)}{\log(t)})} >  g_{y}^{h}(t) \Big)
   \\
   \label{ai}
       &\overset{\ref{1fyh}}\leq     \f{\Phi_y^h(t)}{ f(y) - h  } \pp\Big(   X_t^{(0,\f{g_{y}^{h}(t)}{\log(t)})} >  g_{y}^{h}(t) \Big)  \overset{\ref{keylemma}}\leq    \f{\Phi_y^h(t)}{ f(y) - h  }    \exp\l(   (*)    \r)   t^{-n},
   \\
   \nonumber
   (*)  &\overset{\phantom{\ref{1fyh}}}\lesssim         t      \log(t^n)  \     t^{ n   \f{g_{y}^{h}(t)}{g_{y}^{h}(t)\log(t)}    }    \  \ov\l( \f{g_{y}^{h}(t)}{\log(t)} \r)      \f{g_{y}^{h}(t)}{g_{y}^{h}(t)\log(t)}
   \\
         &  \overset{\phantom{\ref{1fyh}}}= n e^n t \ov\l( \f{g_{y}^{h}(t)}{\log(t)} \r)      .
   \end{align}
     Now,    $\lim_{t\to\infty}   t \ov\l( g_{y}^{h}(t) / \log(t)\r)=0$, uniformly in $h>0$, $y>g(h)$, by Lemma \mbox{\ref{1.1}} and $\l(\ref{condt}\r)$, and thus  $\lim_{t\to\infty}(*)=0$. Then  as $t\to\infty$,  uniformly in $y,h$,

  $(\mbox{\hyperref[ab]{$a$}})   \lesssim t^{-n}   \Phi_y^h(t)     / (f(y)-h  )\leq    t^{-1}\log(t)^{-1-\eps}    \Phi_y^h(t)     / (f(y)-h  ) $,  as required for~$\l(\ref{1'}\r)$.

  To show $(\mbox{\hyperref[ab]{$a$}}) \leq \Phi_y^h(t) u(t) \ov(g(t)) /(f(y)-h) $, we split   up upon the  size of $g(t)$.
 Fix large $M>0$. By $\l(\ref{ai}\r)$, for all $t>0$ such that $g(t) \leq t^M$,
   \(
        (\mbox{\hyperref[ab]{$a$}}) \lesssim    t^{-n}  ,
   \)
      so for   $n$ so that $t^{-n} \leq \ov(g(t))$,
 by Lemma \mbox{\ref{1fyh}},   $(\mbox{\hyperref[ab]{$a$}})\lesssim  \Phi_y^h(t)  u(t) \ov(g(t))  /(f(y)-h)   $   for suitable $u$, as $t\to\infty$, uniformly in $h>0,y>g(h)$, as required for $\l(\ref{2'}\r)$.

   For  all $t>0$ with  $g(t)\geq t^M$,     by   Lemma \hbox{\ref{keylemma}} with $H(t) =  \ov(g(t))^2$,  applying Lemma \mbox{\ref{1.1}}, as $\ov$ is \mbox{\hyperref[rv]{regularly varying}} at $\infty$, uniformly in $y,h$, as $t\to\infty$,
\b{align}
\label{aaa}
(\mbox{\hyperref[ab]{$a$}}) &\overset{\ref{keylemma}}\leq           \exp\l(  (*)    \r)  \ov(g(t))^2,
\\
\nonumber
(*) &\overset{\phantom{\ref{keylemma}}}\lesssim    t           \log\l(\ov(g(t))^{-2}\r)       \f{   }{ }    \ov(g(t))^{-\f{2}{\log(t)}}    \ov\l( \f{g_{y}^{h}(t)}{\log(t)}  \r)         \f{ g_{y}^{h}(t)   }{    g_{y}^{h}(t)       \log(t)     }
\\
\nonumber
  &\overset{\ref{1.1}}\lesssim   t     \ov(g(t))^{-\f{2}{\log(t)}}     \f{       \log\l(\ov(g(t))^{-2}\r) }{  \log(t)     }   \ov\l( \f{g(t)}{\log(t)}  \r).
 \intertext{Now,  note that as $t\to\infty$, for arbitrarily small $\eta>0$,   $ \ov(g(t))^{- 2/\log(t)}  \leq \ov(g(t))^{-\eta}$ and    $\log\l(\ov(g(t))^{-2}\r) \leq \ov(g(t))^{-\eta}$. Since  $\ov$ is  regularly varying at $\infty$ with index $-\alpha$, $\ov(x)=x^{-\alpha} L(x)$, for $L$ \mbox{\hyperref[sv]{slowly varying}} at $\infty$, so as $t\to\infty$,}

  \nonumber
 (*)  &\lesssim      t        \f{       1  }{      \log(t)     }   \ov\l( \f{g(t)}{\log(t)}  \r)   \ov\l( g(t)  \r)^{-2\eta}

 \nonumber

  =      t        \f{       1  }{      \log(t)     }         g(t)^{-(1-2\eta)\alpha}   \log(t)^{\alpha}L\l(  \f{ g(t)}{\log(t)} \r)      L(   g(t) )^{-2\eta}

\\
\nonumber

  &\lesssim  t             g(t)^{-(1-2\eta)\alpha}  L\l(  \f{ g(t)}{\log(t)} \r)      L(   g(t) )^{-2\eta}

     \leq         t             g(t)^{-\f{(1-2\eta)\alpha}{2}}     .
 \end{align}
 Choosing   $M>4/( (1-2\eta )  \alpha)$,  we get    $g(t)^{-(1-2\eta)\alpha/2}\leq    t^{-(1-2\eta)\alpha M/2}\leq t^{-2}$, so  $\lim_{t\to\infty}(*)=0$. By Lemma \mbox{\ref{1fyh}} and $\l(\ref{aaa}\r)$,     $(\mbox{\hyperref[ab]{$a$}})\leq \Phi_y^h(t)  u(t) \ov(g(t)) / (f(y)-h) $   as $t\to\infty$, uniformly in $h>0$, $y>g(h)$, for suitable $u$, as required for $\l(\ref{2'}\r)$.

 \begin{comment}
\n In case (\hbox{\hyperref[case2]{ii}}), we  partition $     \pp\Big(   \oo_{t,X^{(0,g_{y}^{h}(t))}
   }^{ g_{y}^{h} } \Big)$  differently to in case (\hbox{\hyperref[case1]{i}}), which means we have $(a')$ and $(b')$ in place of $(\mbox{\hyperref[ab]{$a$}})$ and $(\mbox{\hyperref[ab]{$b$}})$.  For small   $c\in(0,1)$,
\b{align}
\nonumber
        \pp\l(   \oo_{t,X^{(0,g_{y}^{h}(t))}
   }^{ g_{y}^{h} } \r)  &= \pp\l(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ; \D^{c g_{y}^{h}(t)   }     > t \r)   + \pp\l(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ; \D^{  c g_{y}^{h}(t)   }     \leq t \r)
\\
\label{a'b'}
     &=:(a')+(b').
   \end{align}
    \end{comment}

\begin{comment}
\p{Proof for $(a')$,  Case (\hbox{\hyperref[case2]{ii}})}   \hspace{-10pt} By $\l(\ref{subevent}\r)$ and Lemma \hbox{\ref{keylemma}} with $H(t) \hspace{-2pt}=\hspace{-2pt} \ov(g(t))^2 $,~as~$t~\hspace{-3pt}\to~\hspace{-3pt}\infty$,
   \b{align}
   \label{aaa'}
   (\mbox{\hyperref[a'b']{$a'$}}) &\overset{\l(\ref{subevent}\r)}\leq \pp\l(   X_t^{(0,c g_{y}^{h}(t) )} >  g_{y}^{h}(t) \r)  \overset{\ref{keylemma}}\leq \exp\l(    (*)    \r) \ov(g(t))^2 ,

\\

\nonumber
   (*)   &\overset{\phantom{\l(\ref{subevent}\r)}}\lesssim t    \log\l( \ov(g_{y}^{h}(t))^{-2}   \r)          \ov(g(t))^{-2c}      \f{c   g_{y}^{h}(t)    }{    g_{y}^{h}(t)    }  \ov(g_{y}^{h}(t)) .

\intertext{By Lemma  \mbox{\ref{1.1}}, $\ov(g_{y,h}(t)) \lesssim \ov(g(t))$ as $t\to\infty$, uniformly among $y,h$. So  for arbitrarily small $\eta>0$, as $t\to\infty$, uniformly among $h>0, y>g(h)$,}

\nonumber
(*)  &\overset{\phantom{\l(\ref{subevent}\r)}}\lesssim  t      \ov(g(t))^{1 -\eta -  2c    }.
  \end{align}
 Choosing $c,\eta$ small enough, it follows from  $\l(\ref{cond2}\r)$    that $\limsup_{t\to\infty}(*)=0 $,   so by Lemma  \mbox{\ref{1fyh}} and $\l(\ref{aaa'}\r)$,  $(\mbox{\hyperref[a'b']{$a'$}})\lesssim   \ov(g(t))^2 \leq   \Phi_y^h(t)  u(t) \ov(g(t)) / (f(y)-h)  $  uniformly in $h>0$, $y>g(h)$,  for suitable $u$, as required for $\l(\ref{2'}\r)$ in case (\hbox{\hyperref[case2]{ii}}).

   \end{comment}

   \p{Partitioning (\mbox{\hyperref[ab]{$b$}})
   }
 Now we   partition $(\mbox{\hyperref[ab]{$b$}})$  in case (\hbox{\hyperref[case1]{i}}).   Let $\Delta_m^x$ denote the time of our subordinator's $m$th jump of size larger than $x$, as in $\l(\ref{D}\r)$.      With $\beta$ as in $\l(\ref{condt}\r)$,  for $m>0$ such that $m> \beta/(\beta-1)$ and  $m > 1/(\alpha(\beta-1)) $, for $c>0$ such that  $1    -(   m    -    1) c >0$, and for all $t$ large enough that $1/\log(t)   \leq c$,
   \b{align}
   \nonumber
   (\mbox{\hyperref[ab]{$b$}}) &= \pp  \hspace{1pt}   \Big(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ; \D^{\f{g_{y}^{h}(t)}{\log(t)}   }     \leq t \Big)           \phantom{\Bigg(}
 \\
 \nonumber
   &= \pp \hspace{1pt}  \Big(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ; \D^{\f{g_{y}^{h}(t)}{\log(t)}   }     \leq t   ; \Delta_m^{\f{g_{y}^{h}(t)}{\log(t)}   }     > t     ;   \D^{    c g_{y}^{h}(t)   }     > t   \Big)         \phantom{\Bigg(}
\\
\nonumber
   &+
     \pp  \hspace{1pt}   \Big(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ;\Delta_m^{\f{g_{y}^{h}(t)}{\log(t)}   }     \leq  t        ;   \D^{    c g_{y}^{h}(t)   }     > t    \Big)            \phantom{\Bigg(}
   \\
   \nonumber
     &+
   \pp\l(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ;     \D^{    c g_{y}^{h}(t)   }     \leq t  \r)        \phantom{\Bigg(}
\\
\label{2ABC}
   &=: (2A) + (2B) + (2C).
   \end{align}

   \p{Proof for (\mbox{\hyperref[2ABC]{$2A$}})
   } Disintegrating on the value of $\D^{\f{g_{y}^{h}(t)}{\log(t)}   } $,
   \b{align*}
   (\mbox{\hyperref[2ABC]{$2A$}}) &= \pp \hspace{1pt} \Big(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ; \D^{\f{g_{y}^{h}(t)}{\log(t)}   }     \leq t   ; \Delta_m^{\f{g_{y}^{h}(t)}{\log(t)}   }     > t     ;   \D^{    c g_{y}^{h}(t)   }     > t   \Big)

\\

   &= \int_0^t     \pp\Big(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} }  ; \Delta_m^{\f{g_{y}^{h}(t)}{\log(t)}   }     > t     ;   \D^{    c g_{y}^{h}(t)   }     > t  \ \Big| \      \D^{\f{g_{y}^{h}(t)}{\log(t)}   }      = s  \Big)    \pp\Big(   \D^{\f{g_{y}^{h}(t)}{\log(t)}   }      \in ds  \Big)

 \\

   &\leq  \ov\l(  \f{g_{y}^{h}(t)}{\log(t)}       \r)   \hspace{-1pt}   \int_0^t     \pp\Big(  \oo_{t,X^{(0,c g_{y}^{h}(t))} }^{ g_{y}^{h} }  ; \Delta_m^{\f{g_{y}^{h}(t)}{\log(t)}   }     > t      \ \Big| \      \D^{\f{g_{y}^{h}(t)}{\log(t)}   }      = s  \Big)   ds,

   \end{align*}
  since $\D^{\f{g_{y}^{h}(t)}{\log(t)}   } $ is exponentially distributed. Now, partitioning the event further,
\b{align*}
&     \pp\Big(  \oo_{t,X^{(0,c g_{y}^{h}(t))} }^{ g_{y}^{h} }  ; \Delta_m^{\f{g_{y}^{h}(t)}{\log(t)}   }     > t      \ \Big| \      \D^{\f{g_{y}^{h}(t)}{\log(t)}   }      = s  \Big)
\\

    &=   \hspace{-2pt} \sum_{k=1}^{m-1}   \hspace{-2.5pt}     \pp\Big(  \hspace{-2pt}  \oo_{t,X^{(0,c g_{y}^{h}(t))} }^{ g_{y}^{h} }         \Big|       \D^{\f{g_{y}^{h}(t)}{\log(t)}   }     \hspace{-2pt}       =   \hspace{-2pt}    s  ; \Delta_{k+1}^{\f{g_{y}^{h}(t)}{\log(t)}   }    \hspace{-2pt}       >   \hspace{-2pt}    t     ;  \Delta_k^{\f{g_{y}^{h}(t)}{\log(t)}   }   \hspace{-3pt}    \leq    \hspace{-2pt}    t  \hspace{-1pt}  \Big)  \pp\Big( \hspace{-1pt}   \Delta_{k+1}^{\f{g_{y}^{h}(t)}{\log(t)}   } \hspace{-2pt}     >  \hspace{-2pt}    t     ;  \Delta_k^{\f{g_{y}^{h}(t)}{\log(t)}   }   \hspace{-2pt}   \leq  \hspace{-2pt}    t  \hspace{-1pt}   \Big)

\\

    &\leq    \sum_{k=1}^{m-1}  \pp\Big(  \oo_{t,X^{(0,c g_{y}^{h}(t))} }^{ g_{y}^{h} }        \ \Big| \      \D^{\f{g_{y}^{h}(t)}{\log(t)}   }      = s  ; \Delta_{k+1}^{\f{g_{y}^{h}(t)}{\log(t)}   }     > t     ;  \Delta_k^{\f{g_{y}^{h}(t)}{\log(t)}   } \leq  t  \Big)  ,

  \end{align*}
  and then it follows that

      \b{align}
      \nonumber
   (\mbox{\hyperref[2ABC]{$2A$}}) &\overset{\phantom{\l(\ref{subevent}\r)}}\leq   \hspace{-1pt}       \sum_{k=1}^{m-1}      \ov  \hspace{-1pt}    \l(     \f{g_{y}^{h}(t)}{\log(t)}     \hspace{-1pt}      \r)  \hspace{-3pt}      \int_0^t       \hspace{-2pt}    \pp   \Big(   \hspace{-1pt}   \oo_{ \hspace{-2pt} t,X^{(0, c  g_{y}^{h}(t)   )} }^{ g_{y}^{h} }     \Big|     \D^{\f{g_{y}^{h}(t)}{\log(t)}   }     \hspace{-1.5pt} =    \hspace{-1.5pt}  s    ;  \Delta_{k+1}^{\f{g_{y}^{h}(t)}{\log(t)}   }     \hspace{-1pt}   >  \hspace{-1pt}   t     ;  \Delta_{k}^{\f{g_{y}^{h}(t)}{\log(t)}   }   \hspace{-1pt}   \leq    t    \Big)   \hspace{-1pt}     ds    .

\intertext{Now, by $\l(\ref{subevent}\r)$, given that there are $k$  jumps of size in $[  g_{y}^{h}(t)/\log(t),c  g_{y}^{h}(t)]$,}

\nonumber
 (\mbox{\hyperref[2ABC]{$2A$}})  &\overset{\l(\ref{subevent}\r)}\leq         \sum_{k=1}^{m-1}    \ov  \hspace{-1pt}    \l(        \hspace{-2pt}    \f{g_{y}^{h}(t)}{\log(t)}     \hspace{-2pt}      \r)  \hspace{-4pt}      \int_0^t       \hspace{-5pt}    \pp   \hspace{-0.5pt}     \Big(   \hspace{-2pt}     X_t^{(0, c  g_{y}^{h}(t)   )} \hspace{-2pt}    >    \hspace{-2pt}      g_{y}^{h}(t)   \Big| \hspace{-1pt}      \D^{\f{g_{y}^{h}(t)}{\log(t)}   }     \hspace{-3.5pt} =    \hspace{-2.5pt}  s   ;  \hspace{-1pt}    \Delta_{k+1}^{\f{g_{y}^{h}(t)}{\log(t)}   }     \hspace{-2pt}   >  \hspace{-2pt}   t     ; \hspace{-1pt}   \Delta_{k}^{\f{g_{y}^{h}(t)}{\log(t)}   }   \hspace{-1pt}   \leq  \hspace{-1pt}    t     \hspace{-1.5pt}   \Big)   \hspace{-1pt}     ds

 \\
 \nonumber
 &\overset{\phantom{\l(\ref{subevent}\r)}}\leq        \sum_{k=1}^{m-1}      \ov\l(  \f{g_{y}^{h}(t)}{\log(t)}       \r)      \int_0^t          \pp\Big(       X_{t-s}^{\big(0, \f{g_{y}^{h}(t)}{\log(t)}\big)} > (1-kc) g_{y}^{h}(t)    \Big)   ds

 \\
 \nonumber

   &\overset{\phantom{\l(\ref{subevent}\r)}}\leq      \sum_{k=1}^{m-1}      \ov\l(  \f{g_{y}^{h}(t)}{\log(t)}       \r)       t    \    \pp\Big(    X_t^{\big(0,  \f{g_{y}^{h}(t)}{\log(t)}   \big)}      >     (1-kc) g_{y}^{h}(t)   \Big)
   \\
   \nonumber
   &\overset{\phantom{\l(\ref{subevent}\r)}}\leq   ( m-1)      \ov\l(  \f{g_{y}^{h}(t)}{\log(t)}       \r)       t    \    \pp\Big(    X_t^{\big(0,  \f{g_{y}^{h}(t)}{\log(t)}   \big)}      >     (1-(m-1)c) g_{y}^{h}(t)   \Big)   .

\intertext{Now,  $\lim_{t\to\infty}  t   \ov(  g_{y}^{h}(t)/\log(t)       )   \geq \lim_{t\to\infty}  t   \ov(  (1-1/A)g(t)/\log(t)       )    =0  $ by $\l(\ref{condt} \r)$, uniformly in $h>0,y>g(h)$ by  Lemma \mbox{\ref{1.1}}. Applying  Lemma \mbox{\ref{keylemma}} with $H(t)  =  t^{-2}$, as $g_y^h(t)   \geq   (1-1/A)g(t) $,  uniformly in $h  >  0,y  >  g(h)$, as $t\to\infty$,  }

   \label{2Alabel}
      (\mbox{\hyperref[2ABC]{$2A$}}) &\lesssim \pp\Big(    X^{\big(0,  \f{g_{y}^{h}(t)}{\log(t)}   \big)}      >     (1-(m-1)c) g_{y}^{h}(t)   \Big)  \leq    \exp\l(  (*)  \r)    t^{-2}   ,

  \\
  \nonumber

   (*)&\lesssim    t \log(t^{2})    t^{\f{2}{(1-(m-1)c) \log(t) }}         \ov\l( \f{g_{y}^{h}(t)}{\log(t)} \r)         \f{g_{y}^{h}(t)}{\log(t) (1-(m-1)c) g_{y}^{h}(t) }

 \\
 \nonumber

  &=     \f{2}{1-(m-1)c}    e^{\f{2}{1-(m-1)c}}    t          \ov\l( \f{g_{y}^{h}(t)}{\log(t)} \r)    .

   \intertext{By $\l(\ref{condt}\r)$ and Lemma \mbox{\ref{1.1}}, it follows that $\lim_{t\to\infty}(*)=0$, uniformly among $h  >  0$ and $y  >  g(h)$, so by Lemma \mbox{\ref{1fyh}},    $(\mbox{\hyperref[2ABC]{$2A$}})   \lesssim    t^{-2}    \lesssim        t^{-1}\log(t)^{-1-\eps}    \Phi_y^h(t)     / (f(y)-h  ) $ as $t  \to  \infty$,  uniformly in $h  >  0$, $y  >  g(h)$, as required for $\l(\ref{1'}\r)$.}

   \intertext{Now we    prove   $(\mbox{\hyperref[2ABC]{$2A$}})\leq   \Phi_y^h(t)  u(t) \ov(g(t))  $. Fixing $M>0$, we split   into two subsets of $t>0$.
For all $t>0$ such that $g(t) \leq t^M$,   by $\l(\ref{2Alabel}\r)$ and  Lemma  \mbox{\ref{keylemma}} with~$H(t) =  t^{-n}$, uniformly in $h>0,y>g(h)$,   as $g(t)^{1/M} \leq t\to\infty$,}

\nonumber
  (\mbox{\hyperref[2ABC]{$2A$}})   &\overset{\l(\ref{2Alabel}\r)}\lesssim   \hspace{-2pt}  \pp\l(    X^{(0,  \f{g_{y}^{h}(t)}{\log(t)}   )}      >     (1-(m-1)c) g_{y}^{h}(t)   \r) \leq   \exp\l(  (*) \r)   t^{-n},

 \\
 \nonumber

  (*) &\overset{\phantom{\l(\ref{2Alabel}\r)}}\lesssim  \hspace{-2pt}  t \log(t^n)       e^{\f{n }{1-(m-1)c   }}          \ov\l(  \f{g_{y}^{h}(t)}{   \log(t)}    \r)        \f{g_{y}^{h}(t)}{   \log(t)    (1-(m-1)c)  g_{y}^{h}(t)  }
\lesssim       t \ov\l(\f{g_{y}^{h}(t)}{\log(t)}\r )     \hspace{-2pt}    ,

  \intertext{and by $\l(\ref{condt}\r)$ and Lemma \mbox{\ref{1.1}}, it follows that $\lim_{t\to\infty}(*)=0$, uniformly among $h>0$, $y>g(h)$. Then choosing $n$ large enough that $t^{-n} \leq \ov(g(t))^2$ (this is possible as $\ov$ is \mbox{\hyperref[rv]{regularly varying}} at $\infty$ and $g(t)\leq t^M$),  by Lemma \mbox{\ref{1fyh}}, $(\mbox{\hyperref[2ABC]{$2A$}}) \lesssim  \ov(g(t))^2    \leq  \Phi_y^h(t) u(t) \ov(g(t)) / (f(y)-h) $ as $ g(t)^{1/M} \leq t\to\infty$ for suitable  $u$, uniformly in $h>0,y>g(h)$, as required for $\l(\ref{2'}\r)$.}

  \intertext{For all $t>0$ for which $g(t)\geq t^M  $,   by  $\l(\ref{2Alabel}\r)$ and  Lemma \mbox{\ref{keylemma}} with $H(t) = \ov(g(t))^2  $, uniformly in $h>0,y>g(h)$, as $g(t)^{1/M} \geq t\to\infty$,          }

\label{2aaaaa}
   (\mbox{\hyperref[2ABC]{$2A$}})   &\overset{\l(\ref{2Alabel}\r)}\lesssim    \pp\l(    X^{(0,  \f{g_{y}^{h}(t)}{\log(t)}   )}      >     (1-(m-1)c) g_{y}^{h}(t)   \r) \leq   \exp\l( (*)  \r)      \ov(    g(t))^2,

  \\
  \nonumber

   (*)    &\overset{\phantom{\l(\ref{2Alabel}\r)}}\lesssim   t \log( \ov(g(t))^{-2} )      \ov(g(t))^{ - \f{ 2 }{ (1-(m-1)c)  \log(t)}}            \f{    \ov\big(   \f{g_{y}^{h}(t)}{   \log(t)}   \big)     g_{y}^{h}(t)}{   \log(t)  (1-(m-1)c) g_{y}^{h}(t) }

  \\
  \nonumber

   &\overset{\phantom{\l(\ref{2Alabel}\r)}}\lesssim  t \log( \ov(g(t))^{-2} )    \ov(g(t))^{ - \f{ 2 }{ (1-(m-1)c)  \log(t)}}       \f{    \ov\big(   \f{g_{y}^{h}(t)}{   \log(t)}   \big)    }{   \log(t)   }     .

   \intertext{As  $\ov$ is  regularly varying at $\infty$, $\ov(x)=x^{-\alpha}L(x)$, for $L$ \mbox{\hyperref[sv]{slowly varying}}  at $\infty$. For each $\eta>0$, as $t\to\infty$,    $  \ov(g(t))^{    -   2 / ((1-(m-1)c)  \log(t)) }     \leq     g(t)^{\alpha\eta}   L(g(t))^\eta   $     and      $ \log( \ov(g(t))^{-2} )  \hspace{-2.5pt}   \leq   \hspace{-2.5pt} g(t)^{\alpha\eta}   L(g(t))^\eta$.     Then by      Lemma     \mbox{\ref{1.1}},  uniformly~in $h    \hspace{-2.5pt}  >    \hspace{-2.5pt}  0,   y      \hspace{-2pt}  >  \hspace{-2pt}g(h)$, as $g(t)^{1/M}  \geq  t        \to    \infty$,     }

  \nonumber
   (*) &\overset{\phantom{\ref{1.1}}}\lesssim  t    g(t)^{2\alpha\eta}   L(g(t))^{2\eta}         \f{    \ov\big(   \f{g_{y}^{h}(t)}{   \log(t)}   \big)   }{   \log(t)   }

  \hspace{80pt} \overset{\ref{1.1}}\lesssim     t    g(t)^{2\alpha\eta}   L(g(t))^{2\eta}         \f{    \ov\big(   \f{g(t)}{   \log(t)}   \big)   }{   \log(t)   }

    \\
    \nonumber
    &\overset{\phantom{\ref{1.1}}}=  t    g(t)^{-(1-2\eta)\alpha}   L(g(t))^{2\eta}         \log(t)^{\alpha-1} L\l(\f{g(t)}{\log(t)}\r)

    \overset{\phantom{\ref{1.1}}}\lesssim     t    g(t)^{-\f{(1-2\eta)\alpha}{2}}           .
     \end{align}

\n Now, $g(t)\geq t^M$,  so  $tg(t)^{-(1-2\eta)\alpha} \leq t^{1-(1-2\eta)\alpha M /2}$,  so taking    $M >  2/( (1-2\eta) \alpha)  $,    $\lim_{t\to\infty}(*)=0$, uniformly in $y,h$. Then by $\l(\ref{2aaaaa}\r)$ and Lemma \mbox{\ref{1fyh}},    it follows that as $t\to\infty$,  $(\mbox{\hyperref[2ABC]{$2A$}}) \leq  \ov(g(t))^2    \lesssim  \Phi_y^h(t) u(t) \ov(g(t))  / (f(y)-h) $  for suitable $u$,   uniformly in $h>0,y>g(h)$,  as required for $\l(\ref{2'}\r)$.

 \p{Proof for (\mbox{\hyperref[2ABC]{$2B$}})}   Disintegrating on the value of $ \Delta_1^{\f{g_{y}^{h}(t) }{\log(t)}  }   $, which   is exponentially distributed with parameter $\ov\l(g_{y}^{h}(t) /\log(t)\r)$,
 \b{align}
 \nonumber
 (\mbox{\hyperref[2ABC]{$2B$}}) &=    \pp\Big(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ;\Delta_m^{\f{g_{y}^{h}(t) }{\log(t)}  }     \leq  t        ;   \D^{    c g_{y}^{h}(t)   }     > t    \Big)
\\
\nonumber
 &\leq      \ov\l(\f{g_{y}^{h}(t) }{\log(t)} \r)  \int_0^t          \pp\Big(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ;\Delta_m^{\f{g_{y}^{h}(t) }{\log(t)}   }     \leq  t        ; \D^{    c g_{y}^{h}(t)   }     > t  \  \big| \    \Delta_1^{\f{g_{y}^{h}(t) }{\log(t)}  }   =s  \Big)      ds
\\
\nonumber
 &\leq    \ov\l(\f{g_{y}^{h}(t) }{\log(t)} \r) \int_0^t          \pp\Big(  \oo_{s }^{ g_{y}^{h} } ;\Delta_m^{\f{g_{y}^{h}(t) }{\log(t)}  }     \leq  t        \    \big|  \    \Delta_1^{\f{g_{y}^{h}(t) }{\log(t)}  }   =s  \Big)     ds   .

 \intertext{By  $\l(\ref{s-}\r)$,    $\cal{O}_{s}^{g_y^h} = \cal{O}_{s-  }^{g_y^h}  $. Given $\Delta_1^{ \f{g_y^h(t)}{\log(t) }} =s$,        $ \cal{O}_{s-}^{g_y^h} $  is independent of $\Delta_1^{ \f{g_{y}^{h}(t) }{\log(t)}}   $~and~$\Delta_m^{ \f{g_{y}^{h}(t) }{\log(t)}} \hspace{-4pt} $,~so}

\nonumber
  (\mbox{\hyperref[2ABC]{$2B$}})  &\leq     \ov\l(\f{g_{y}^{h}(t) }{\log(t)} \r) \int_0^t          \pp\Big(  \oo_{s  }^{ g_{y}^{h} }  \Big)   \pp\Big(     \Delta_m^{\f{g_{y}^{h}(t) }{\log(t)} }     \leq  t             \   \big|  \   \Delta_1^{ \f{g_{y}^{h}(t) }{\log(t)} }   =s   \Big)       ds
\\
\nonumber
  &\leq     \ov\l(\f{g_{y}^{h}(t) }{\log(t)} \r)     \pp\Big(     \Delta_{m-1}^{\f{g_{y}^{h}(t) }{\log(t)}   }     \leq  t    \Big)        \int_0^t          \pp\Big(  \oo_{s }^{ g_{y}^{h} }  \Big)   ds.

\\
\intertext{Now,  since $ \Delta_1^{\f{g_{y}^{h}(t) }{\log(t)}  }   $  is exponentially distributed with parameter $\ov\l(g_{y}^{h}(t) /\log(t)\r)$,
\[
   \pp\Big(     \Delta_{m-1}^{\f{g_{y}^{h}(t) }{\log(t)}   }     \leq  t    \Big)        \leq   \pp\Big(     \Delta_{1}^{  \f{g_{y}^{h}(t) }{\log(t)}   }     \leq  t    \Big)^{m-1}     \hspace{-0.3cm} =    \Big(     1 - e^{- t \ov\big(  \f{g_{y}^{h}(t) }{\log(t)}    \big)     } \Big)^{m-1} \hspace{-0.3cm}     \leq        t^{m-1} \ov\l(  \f{g_{y}^{h}(t) }{\log(t)}  \r)^{  \hspace{-0.15cm}  m-1}
  \hspace{-0.15cm} ,\]
so recalling the notation in $\l(\ref{phiyh}\r)$, by Lemma \mbox{\ref{1.1}}, uniformly in $y,h$ as $t\to\infty$,}

  \\ \label{2Bstep}

 (\mbox{\hyperref[2ABC]{$2B$}})
 &\overset{\phantom{\ref{1.1}}}\leq        \ov\l(\f{g_{y}^{h}(t) }{\log(t)} \r)^{m}   t^{m-1}   \Phi_y^h(t)

 \overset{\ref{1.1}}\lesssim     \ov\l(\f{g(t) }{\log(t)} \r)^m t^{m-1}  \Phi_y^h(t).
  \end{align}

 Recall $\ov(x)=x^{-\alpha} L(x)$ for $L$  \mbox{\hyperref[sv]{slowly varying}}  at $\infty$, so by Potter's theorem \cite[Theorem 1.5.6]{bgt89}, for arbitrarily small $\delta>0$, as $t\to\infty$,
 \b{equation} \label{log1}
   \ov\l(\f{g(t) }{\log(t)} \r)    = \log(t)^{\alpha} g(t)^{-\alpha} L(g(t))  \f{   L\l(\f{g(t) }{\log(t)} \r)  }{ L(g(t))  }  \lesssim   \log(t)^{\alpha + \delta}  \ov(g(t)).
 \end{equation}
 Similarly, defining   $g_\beta(t):=g(t)/\log(t)^\beta$, for $\beta>(1+\alpha)/(2\alpha+\alpha^2)$ as in $\l(\ref{condt}\r)$,
 \b{equation}  \label{log2}
    \ov\l(\f{g(t) }{\log(t)} \r)   =   \log(t)^{\alpha} g(t)^{-\alpha}   L\l(  g_\beta(t)   \r)  \f{   L\l(\f{g(t) }{\log(t)} \r)  }{ L(g_\beta(t))  }  \lesssim     \log(t)^{\alpha(1-\beta) + \delta\beta}\ov(g_\beta(t)).
 \end{equation}

Applying $\l(\ref{log1}\r)$ to $\ov(g(t))$ and $\l(\ref{log2}\r)$ to $\ov(g(t))^{m-1}$, then by $\l(\ref{condt}\r)$, as $t\to\infty$,

 \begin{align}
 \nonumber
  (\mbox{\hyperref[2ABC]{$2B$}})
 &\overset{\phantom{\ref{1.1}}}\lesssim  \log(t)^{ m\alpha + m \delta  - (m-1)\alpha \beta}      \ov(g(t)) \ov(g_\beta(t))^{m-1}    t^{m-1}    \Phi_y^h(t)
 \\
 \nonumber
 &\overset{(\ref{condt})}\leq    \log(t)^{ m\alpha + m \delta  - (m-1)\alpha \beta}      \ov(g(t))  \Phi_y^h(t).

 \intertext{Now,   for $\beta$ as in $\l(\ref{condt}\r)$,  $m>\beta/(\beta-1)$, so  $m\alpha -   (m-1)\alpha\beta <0$,   choosing $\delta>0$ small enough, we conclude that for suitable  $u$,  $ (\mbox{\hyperref[2ABC]{$2B$}})\leq  u(t) \ov(g(t)) \Phi_y^h(t)   $   as $t\to\infty$, uniformly among $h>0,y>g(h)$,  as required for $\l(\ref{2'}\r)$.}

  \intertext{Now we prove  that $(\mbox{\hyperref[2ABC]{$2B$}})\lesssim        t^{-1}\log(t)^{-1-\eps}    \Phi_y^h(t)       $. With $g_\beta(t)=g(t)/\log(t)^\beta$,  by $\l(\ref{2Bstep}\r)$ and $\l(\ref{log2}\r)$,   for arbitrarily small $\d>0$, as $t\to\infty$, uniformly in $y,h$,}

\nonumber
(\mbox{\hyperref[2ABC]{$2B$}}) &\leq        \ov\l(\f{g(t) }{\log(t)} \r)^m t^{m-1}  \Phi_y^h(t)

\lesssim     \log(t)^{ m \alpha (1-\beta) + m \delta }   \ov(g_\beta(t))^m t^{m-1}  \Phi_y^h(t) .
 \end{align}

By $\l(\ref{condt}\r)$, $\ov(g_\beta(t))\leq t^{-1}$, and so
\(
(\mbox{\hyperref[2ABC]{$2B$}})\lesssim    t^{-1} \log(t)^{-m\alpha(\beta-1) + m\delta (\beta-1)}  \Phi_y^h(t)
\) as $t\to\infty$.
 Finally,  $m\alpha( 1- \beta) <-1$, so  choosing $\d$  small enough,   we conclude that for some $\eps>0$,  $(\mbox{\hyperref[2ABC]{$2B$}})\lesssim         t^{-1}\log(t)^{-1-\eps}    \Phi_y^h(t)     / (f(y)-h  ) $ as $t\to\infty$,  uniformly in $h>0$, $y>g(h)$, as required for $\l(\ref{1'}\r)$.

\p{Partitioning $(\mbox{\hyperref[2ABC]{$2C$}})
$}

  Defining $p(s) := \log(t)^{-\gamma}   $ for  $\gamma:= (1-\alpha)/(2+\alpha)$, denote $p^*(s):=1-p(s)$, and let $\Delta_2^{x}$ denote the time of our subordinator's second jump of size  at least $x$. Then we partition:
   \b{align}
   \nonumber
     (\mbox{\hyperref[2ABC]{$2C$}}) &=    \pp\l(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ;     \D^{    c g_{y}^{h}(t)   }     \leq t  \r)
   \\
   \nonumber
     &=  \pp\l(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ;     \D^{    c g_{y}^{h}(t)   }     \leq t       ;    \D^{    p^*(t) g_{y}^{h}(t)   }     \leq t             \r)
    \\
    \nonumber
     &+  \pp\l(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ;     \Delta_2^{    c g_{y}^{h}(t)   }     \leq t       ;      \D^{    p^*(t) g_{y}^{h}(t)   }     > t          \r)
   \\
   \nonumber
     &+  \pp\l(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ;     \D^{    c g_{y}^{h}(t)   }     \leq t        ;     \Delta_2^{    c g_{y}^{h}(t)   }     > t ;         \D^{    p^*(t) g_{y}^{h}(t)   }     > t               \r)
  \\
  \label{2Cabc}
   &=:  (2Ca) + (2Cb) + (2Cc).
   \end{align}

  \p{Proof for (\mbox{\hyperref[2Cabc]{$2Ca$}})}  As $c\in(0,1)$ is fixed,   $c < 1-\log(t)^{-\gamma}$ as $t\to\infty$,  and so
    \b{align}
    \nonumber
    (\mbox{\hyperref[2Cabc]{$2Ca$}})
   &=     \pp\l(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ;         \D^{    p^*(t) g_{y}^{h}(t)   }     \leq t             \r).

 \intertext{Disintegrating on the value of $   \D^{    p^*(t) g_{y}^{h}(t)   }  $, by $\l(\ref{s-}\r)$,}

\nonumber
 (\mbox{\hyperref[2Cabc]{$2Ca$}}) &\overset{\phantom{\l(\ref{s-}\r)}}\leq   \l[   \ov\l( p^*(t) g_{y}^{h}(t)  \r)    - \ov\l( g_{y}^{h}(t)    \r)   \r]   \int_0^t    \pp\l(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } |        \D^{    p^*(t) g_{y}^{h}(t)   }     =s             \r)      ds

\\

\nonumber
 &\overset{\l(\ref{s-}\r)}\leq    \l[   \ov\l( p^*(t) g_{y}^{h}(t)  \r)    - \ov\l( g_{y}^{h}(t)    \r)   \r]   \int_0^t    \pp\l(  \oo_{s,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} }       \r)      ds

\\

\nonumber
   &\overset{\phantom{\l(\ref{s-}\r)}}\leq  \l[   \ov\l( p^*(t) g_{y}^{h}(t)  \r)    - \ov\l( g_{y}^{h}(t)    \r)   \r]  \Phi_y^h(t).

\intertext{By Lemma  \hbox{\ref{1.1}}, $\ov(g_{y}^{h}(t))\lesssim \ov(g(t))$  uniformly across $y, h$, so we can write}

  (\mbox{\hyperref[2Cabc]{$2Ca$}}) &\leq       \ov\l( g_{y}^{h}(t)  \r)  \l(    \f{    \ov\l( p^*(t) g_{y}^{h}(t)  \r)  }{ \ov\l( g_{y}^{h}(t)  \r)   }     -1       \r)  \Phi_y^h(t)

    \label{2Cabit}

    \lesssim \ov\l( g(t)  \r)  \l(    \f{    \ov\l( p^*(t) g_{y}^{h}(t)  \r)  }{ \ov\l( g_{y}^{h}(t)  \r)   }     -1       \r)  \Phi_y^h(t) .

 \intertext{As $\lim_{t\to\infty}p^*(t)   =  1$ and $\ov$ is  \mbox{\hyperref[crv]{CRV}} at $\infty$,
 \(
    \lim_{t\to\infty}      \ov( p^*(t) g_{y}^{h}(t)  )    / \ov( g_{y}^{h}(t)  )        =  1.
 \)
 Now, $g_y^h(t) \gtrsim g(t)$ uniformly in $y,h$ by Lemma \mbox{\ref{1.1}},  so by $\l(\ref{2Cabit}\r)$,  $(\mbox{\hyperref[2Cabc]{$2Ca$}})\leq    \Phi_y^h(t)  u(t) \ov(g(t))   $ for suitable $u$,  uniformly in $h  >  0$, $y  >  g(h)$,  as~required~for~$\l(\ref{2'}\r)$.}

\intertext{Now we prove     $(\mbox{\hyperref[2Cabc]{$2Ca$}})\lesssim    t^{-1} \log(t)^{-1-\eps}   \Phi_y^h(t)  $.

 As $\ov$ is \mbox{\hyperref[rv]{regularly varying}}   at $\infty$,  $\ov(x)=x^{-\alpha} L(x)$ for $L$  \mbox{\hyperref[sv]{slowly varying}}~at~$\infty$, and by $\l(\ref{2Cabit}\r)$, as $t\to\infty$,    uniformly in $h>0,y>g(h)$, }

\nonumber
 (\mbox{\hyperref[2Cabc]{$2Ca$}}) &\lesssim    \ov\l( g(t)  \r)  \l(   \hspace{-2pt}  \f{    \ov\l( p^*(t) g_{y}^{h}(t)  \r)  }{ \ov\l( g_{y}^{h}(t)  \r)   }     -1    \hspace{-3pt}    \r)  \hspace{-3pt}  \Phi_y^h(t)

    =    \ov\l( g(t)  \r)  \l(   \hspace{-3pt}   \f{ p^*(t)^{-\alpha}    L(  p^*(t)g_{y}^{h}(t))       }{      L(  g_{y}^{h}(t))} -1    \hspace{-3pt}    \r)   \hspace{-2pt} \Phi_y^h(t)   .

 \intertext{By Lemma  \hbox{\ref{1.1}}, $ g_{y}^{h}(t)\gtrsim g(t)$  uniformly in $y>g(h), h>0$, as $t\to\infty$. By Potter's theorem  \cite[Theorem 1.5.6]{bgt89},
  for    arbitrarily small   $\d  >  0$, uniformly in $y  >  g(h), h  >  0$,~as~$t  \to  \infty$, }

\nonumber
 (\mbox{\hyperref[2Cabc]{$2Ca$}}) &\lesssim    \l(   p^*(t)^{-\alpha - \d}          - 1  \r)     \ov( g (t)) \Phi_y^h(t)
\\
\nonumber
 &=     \l(   \l(   1- \log(t)^{-\gamma}  \r)^{-\alpha - \d}          - 1  \r)     \ov( g (t))  \Phi_y^h(t)

\leq   \f{  \log(t)^{-\gamma} }{   1- \log(t)^{-\gamma}}        \ov( g(t)) \Phi_y^h(t) .

 \intertext{For $\beta$ as in $\l(\ref{condt}\r)$, $g_\beta(t):=g(t)/\log(t)^\beta$,  by  Potter's theorem and     $\l(\ref{condt}\r)$,  for   arbitrarily small $\tau>0$,  \(
 \ov(g(t))\lesssim    \ov( g_{\beta}(t))  \log(t)^{-\alpha \beta  + \beta \tau}  \leq    t^{-1} \log(t)^{-\alpha \beta  + \beta \tau}
\)   as $t\to\infty$.
Then observing that   $ \lim_{t\to\infty}(1- \log(t)^{-\gamma} )=1$, it follows that uniformly in $y,h$, as $t\to\infty$,}

\nonumber
 (\mbox{\hyperref[2Cabc]{$2Ca$}}) &\lesssim     t^{-1}     \log(t)^{-\gamma -\alpha \beta  + \beta \tau} \Phi_y^h(t).
\end{align}
Since $\gamma = (1-\alpha)/(2+\alpha) >1-\alpha\beta$, we may choose $\tau $ sufficiently  small    that $-\gamma -\alpha \beta  + \beta \tau<-1-\eps<-1 $.  Then it follows that
 $(\mbox{\hyperref[2Cabc]{$2Ca$}})\lesssim           t^{-1}\log(t)^{-1-\eps}    \Phi_y^h(t)    $ as $t\to\infty$,  uniformly among $h>0$, $y>g(h)$, as required for $\l(\ref{1'}\r)$.

  \p{Proof for (\mbox{\hyperref[2Cabc]{$2Cb$}})}
  Disintegrating on  the value of $\D^{c g_{y}^{h}(t) }$,
  by~$\l(\ref{s-}\r)$,
\begin{align}
  \nonumber
(\mbox{\hyperref[2Cabc]{$2Cb$}}) &\overset{\phantom{\l(\ref{s-}\r)}}=     \pp\l(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ;     \Delta_2^{    c g_{y}^{h}(t)   }     \leq t       ;      \D^{    p^*(t) g_{y}^{h}(t)   }     > t          \r)
\\
  \nonumber
  &\overset{\phantom{\l(\ref{s-}\r)}}\leq   \ov\hspace{-1pt}\l(   c g_{y}^{h}(t)    \r) \hspace{-3pt}\int_0^t   \hspace{-4pt}   \pp   \hspace{-2pt} \l(   \hspace{-2pt}  \oo_{ \hspace{-1pt} t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ;     \Delta_2^{    c g_{y}^{h}(t)   }  \hspace{-3pt}   \leq \hspace{-1pt} t       ;    \hspace{-1pt}  \D^{    p^*(t) g_{y}^{h}(t)   } \hspace{-3pt}    > \hspace{-3pt} t    |   \D^{   c g_{y}^{h}(t)   }     \hspace{-2pt}  =\hspace{-2pt}s \hspace{-1pt} \r)  \hspace{-2pt} ds
\\
  \nonumber
  &\overset{\phantom{\l(\ref{s-}\r)}}\leq       \ov  \hspace{-1pt} \l(   c g_{y}^{h}(t)    \r)  \hspace{-3pt}  \int_0^t   \hspace{-4pt}  \pp \hspace{-2pt} \l(     \hspace{-2pt} \oo_{ \hspace{-1pt} s,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} }    |   \D^{   c g_{y}^{h}(t)   }    \hspace{-2pt}   =\hspace{-2pt}s \hspace{-1pt} \r)    \hspace{-1pt}\pp \hspace{-1pt} \l(   \hspace{-2pt}     \Delta_2^{       c g_{y}^{h}(t)   }   \hspace{-2pt}  \leq \hspace{-2pt} t       |      \D^{      c g_{y}^{h}(t)   }    \hspace{-2pt}   =  \hspace{-2pt}  s    \hspace{-1pt}   \r)  \hspace{-2pt} ds
\\
  \nonumber
  &\overset{\l(\ref{s-}\r)}\leq    \ov\l(      c g_{y}^{h}(t)    \r)    \pp\l(     \D^{      c g_{y}^{h}(t)   }     \leq t     \r)  \int_0^t   \pp\l(    \oo_{s  }^{ g_{y}^{h} } \r)ds.

  \intertext{Recall for $L$    \mbox{\hyperref[sv]{slowly varying}}  at $\infty$, $\ov(x)   =x^{-\alpha} L(x)$.   Observe    $ \pp\big(      \D^{      c g_{y}^{h}(t)   }      \leq    t   \big)       =     1    -     e^{-t \ov(   c g_{y}^{h}(t)   )  }       \leq       t \ov(   c g_{y}^{h}(t))$,  so by   $\l(\ref{phiyh}\r)$ and  Lemma \mbox{\ref{1.1}}, uniformly in  $h  >  0,y    >    g(h)$,~as~$ t    \to    \infty$,}

   (\mbox{\hyperref[2Cabc]{$2Cb$}}) &\overset{\l(\ref{phiyh}\r)}\leq  \ov\l(      c g_{y}^{h}(t)    \r)^2   t       \Phi_y^h(t)

   \label{2cb}

     \overset{\ref{1.1}}\lesssim     \ov(g(t))^2 t   \Phi_y^h(t)

  \overset{\phantom{\ref{1.1}}}=   g(t)^{-2\alpha}  L(g(t))^2   t \Phi_y^h(t)
  \\
  \nonumber
   &\overset{\phantom{\ref{1.1}}}=    \log(t)^{ -2\alpha \beta }     \l(\f{g(t)}{\log(t)^{\beta}}\r)^{-2\alpha} L\l(\f{g(t)}{\log(t)^{\beta}}\r)^2     \f{ L(g(t))^2  }{ L\l(\f{g(t)}{\log(t)^{\beta}}\r)^2 }  t \Phi_y^h(t)
 \\
  \nonumber
   &\overset{\phantom{\ref{1.1}}}=     \log(t)^{ -2\alpha \beta }    \ov\l(\f{g(t)}{\log(t)^{\beta}}   \r)^2         \f{ L(g(t))^2  }{ L\l(\f{g(t)}{\log(t)^{\beta}}\r)^2 }  t \Phi_y^h(t) .

   \intertext{By Potter's theorem  \cite[Theorem 1.5.6]{bgt89}, for arbitrarily small $\d  >  0$, as~$t  \to  \infty$,}

  \nonumber
     (\mbox{\hyperref[2Cabc]{$2Cb$}}) &\overset{\phantom{\ref{1.1}}}\lesssim \log(t)^{ -2\alpha \beta }    \ov\l(\f{g(t)}{\log(t)^{\beta}}   \r)^2       \log(t)^{2 \beta\d}         t \Phi_y^h(t) .

   \intertext{It follows by $\l(\ref{condt}\r)$ that       $ \ov\l(g(t)   /   \log(t)^{\beta}   \r)^2 \lesssim  t^{-2}$ as $t\to\infty$, and hence}

  \nonumber
   (\mbox{\hyperref[2Cabc]{$2Cb$}}) &\overset{\phantom{\ref{1.1}}}\lesssim  \log(t)^{ -2\alpha \beta }         \log(t)^{2 \beta\d}         t^{-1} \Phi_y^h(t)
   .

   \intertext{Now,    $2\alpha\beta > 1$ by $\l(\ref{condt}\r)$. Taking $\d$  small enough that $2 \alpha \beta - 2   \beta\d \geq 1+\eps>1$, it follows that $(\mbox{\hyperref[2Cabc]{$2Cb$}})    \lesssim       t^{-1}     \log(t)^{-1-\eps} \Phi_y^h(t) $, uniformly in $y,h$, as required~for~$\l(\ref{1'}\r)$.}

    \intertext{Now we show   $(\mbox{\hyperref[2Cabc]{$2Cb$}})\leq \Phi_y^h(t) u(t) \ov(g(t))  $.
    By $\l(\ref{2cb}\r)$, since   $\lim_{t\to\infty} t\ov(g(t))=0$ by $\l(\ref{condt}\r)$, uniformly in $h>0,y>g(h)$, as $t\to\infty$,}

  \nonumber
   (\mbox{\hyperref[2Cabc]{$2Cb$}}) &\lesssim    \ov(g(t))^2 t   \Phi_y^h(t) = o(1) \times  \ov(g(t))   \Phi_y^h(t) ,

   \end{align}
  so $(\mbox{\hyperref[2Cabc]{$2Cb$}}) \hspace{-2pt} \leq \hspace{-2pt}  \Phi_y^h(t) u(t) \ov(g(t))  $   for suitable $u$, uniformly in $y,h$, as required~for~$\l(\ref{2'}\r)$.

\p{Partitioning (\mbox{\hyperref[2Cabc]{$2Cc$}})} By the definition $\l(\ref{truncoo}\r)$, disintegrating on the value of  $  \D^{       c g_{y}^{h}(t)   }$,   by $\l(\ref{s-}\r)$ and Lemma \mbox{\ref{1.1}}, uniformly in $h>0,y>g(h)$, as $t\to\infty$,
\b{align}
\nonumber
(\mbox{\hyperref[2Cabc]{$2Cc$}}) &\overset{\phantom{\ref{1.1}}}=  \pp\l(  \oo_{t,X^{(0,g_{y}^{h}(t))} }^{ g_{y}^{h} } ;     \D^{       c g_{y}^{h}(t)   }     \leq t        ;     \Delta_2^{       c g_{y}^{h}(t)   }     > t ;         \D^{    p^*(t) g_{y}^{h}(t)   }     > t               \r)

\\
\nonumber
    &\overset{\phantom{\ref{1.1}}}\leq      \ov(       c g_{y}^{h}(  t  )    )  \hspace{-2pt}    \int_0^t \hspace{-3pt}   \pp    \Big(  \hspace{-2pt}  \oo_{    t,X^{(0,  g_{y}^{h}(t))} }^{ g_{y}^{h} }\hspace{-1pt}  ;    \hspace{-1pt}  \Delta_2^{       c g_{y}^{h}(t)   }\hspace{-1pt}      >\hspace{-1pt}  t      ;       \D^{    p^*(t) g_{y}^{h}(t)   } \hspace{-1pt}     > \hspace{-1pt}   t         \big|  \hspace{-1pt}    \D^{       c g_{y}^{h}(t)   }     =  s \hspace{-1pt}    \Big) \hspace{-1pt}     ds
\\
\nonumber
&\overset{\hspace{-2pt}\l(\ref{s-}\r)\hspace{-2pt}}=   \ov(    c g_{y}^{h}(  t  )   ) \hspace{-3pt}  \int_0^t  \hspace{-3pt}   \pp \hspace{-0.1pt}  \Big(  \hspace{-2pt}   \oo_{ \hspace{-1pt}  s,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  \hspace{-1pt}  ; \hspace{-1pt}    \hat{X}_{t-s}^{(0,   c g_{y}^{h}(t))}    \hspace{-3pt}  + \hspace{-2pt}   X_{s-}^{(0,   c g_{y}^{h}(t))}    \hspace{-1pt} >\hspace{-1pt}     (\hspace{-1pt} 1\hspace{-1pt} -\hspace{-0.5pt} p^*(\hspace{-0.5pt} t \hspace{-0.5pt} )\hspace{-0.5pt} )g_{y}^{h}(\hspace{-0.5pt} t\hspace{-0.5pt} )        \hspace{-1pt}    \Big)\hspace{-1pt} ds
\\
\nonumber
&\overset{\ref{1.1}}\lesssim   \ov(    g(  t  ) ) \hspace{-2pt}  \int_0^t  \hspace{-3pt}   \pp \hspace{-0.1pt}  \Big(    \oo_{ \hspace{-1pt}  s,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  \hspace{-1pt}  ; \hspace{-1pt}    \hat{X}_{t-s}^{(0,   c g_{y}^{h}(t))}    \hspace{-3pt}  + \hspace{-2pt}   X_{s-}^{(0,   c g_{y}^{h}(t))}    \hspace{-1pt} >\hspace{-1pt}     (\hspace{-1pt} 1\hspace{-1pt} -\hspace{-0.5pt} p^*(\hspace{-0.5pt} t \hspace{-0.5pt} )\hspace{-0.5pt} )g_{y}^{h}(\hspace{-0.5pt} t\hspace{-0.5pt} )        \hspace{-1pt}    \Big)\hspace{-1pt} ds,

\intertext{where   $\hat{X}$ is an independent copy of $X$, and we use that the jump at time $s$ has size at most $p^*(t) g_y^h(t)$. Recall that $1-p^*(t) = \log(t)^{-\gamma}$. Then partitioning according to the event $ \big\{X_{s-}^{(0,   c g_{y}^{h}(t))}    >   g_{y}^{h}(t)/(2\log(t)^\gamma)    \big\}$ and its complement,}

\nonumber
(\mbox{\hyperref[2Cabc]{$2Cc$}})&\lesssim       \ov(g(t)) \int_0^t  \pp\l(  \oo_{s,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  \ ; \   \hat{X}_{t-s}^{(0,   c g_{y}^{h}(t))}       >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }           \r)   ds
\\
\nonumber
&+   \ov(g(t)) \int_0^t  \pp\l(  \oo_{s,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  \ ; \      X_{s-}^{(0,   c g_{y}^{h}(t))}    >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }   \r)   ds
\\
\label{SS*}
&=:       (S) + (S^*).
\end{align}
Next we will bound $(S)$, then later we will split up $(S^*)$ into more pieces.

  \p{Proof for (\mbox{\hyperref[SS*]{$S$}})}
  As $\hat{X}$ is an  independent copy of $X$, we can write
  \b{align}
  (\mbox{\hyperref[SS*]{$S$}})

\nonumber
  &=      \ov(g(t)) \int_0^t  \pp\l(  \oo_{s,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  \r)  \pp\l(   X_{t-s}^{(0,   c g_{y}^{h}(t))}       >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }           \r)   ds
\\
\label{S0}
  &\leq   \ov(g(t))          \pp\l( X_{t}^{(0,   c g_{y}^{h}(t))}       >   \f{ g_{y}^{h}(t)     }{   2 \log(t)^\gamma  }           \r)     \Phi_y^h(t).
 \intertext{Since $\ov$ is \mbox{\hyperref[rv]{regularly varying}} at $\infty$,   applying Potter's theorem  \cite[Theorem 1.5.6]{bgt89} to $\ov(g(t))$,    for $\beta$ as in $\l(\ref{condt}\r)$ and for arbitrarily small $\tau>0$, as $t\to\infty$,}

\label{S1}
 (\mbox{\hyperref[SS*]{$S$}}) &\lesssim      t^{-1}   \log(t)^{-\alpha \beta + \tau \beta     }    \pp\l( X_{t}^{(0,   c g_{y}^{h}(t))}       >   \f{ g_{y}^{h}(t)     }{   2 \log(t)^\gamma  }           \r)     \Phi_y^h(t).

\end{align}
By  Lemma \mbox{\ref{keylemma}} with $H(t)=\log(t)^{-1 + \alpha \beta  - 2 \tau \beta}   $,  uniformly in $h>0,y>g(h)$ by Lemma \mbox{\ref{1.1}}, as $  t\to\infty$,
  \b{equation}
  \label{S2}
         \pp\l( X_{t}^{(0,   c g_{y}^{h}(t))}       >   \f{ g_{y}^{h}(t)     }{   2 \log(t)^\gamma  }           \r)     \leq \exp\l(( \ast ) \r)
\log(t)^{-1 + \alpha \beta  - 2 \tau \beta}, \e{equation}

\[
  (*)\lesssim   t  \log\l(    \log(t)^{ 1-\alpha\beta + 2\beta \tau    }     \r) \log(t)^{   ( 1-\alpha\beta + 2\beta \tau)  \f{     c   }{  2\log(t)^\gamma  }    }      \ov(   c g_{y}^{h}(t)) \f{     c   }{  2\log(t)^\gamma  }  .
  \]

 Since $\gamma>0$, for all large enough $t$,   $ \log\l(    \log(t)^{ 1-\alpha\beta + 2\beta \tau    }     \r) \leq \log(t)^{\gamma/3}$ and

\[
   \log(t)^{   ( 1-\alpha\beta + 2\beta \tau)  \f{     c   }{  2\log(t)^\gamma  }    }   \leq     \log(t)^{\f{\gamma}{3}}.
\]

  By $\l(\ref{condt}\r)$,  $\lim_{t\to\infty} t\ov(   c g_{y}^{h}(t))=0$, uniformly in $y,h$  by Lemma \mbox{\ref{1.1}},   so as $t\to\infty$,

\[
  (*) \lesssim \f{  \log(t)^{ \f{\gamma}{3} + \f{\gamma}{3}  }    }{     \log(t)^\gamma}    =       \log(t)^{-\f{\gamma}{3}}     ,
\]

 and hence $\lim_{t\to\infty}(*)=0$. Then by $\l(\ref{S1}\r)$ and $\l(\ref{S2}\r)$, choosing $\tau$ small enough,

\[
  (\mbox{\hyperref[SS*]{$S$}})\lesssim     \f{\Phi_y^h(t)}{t   \log(t)^{1+  \tau \beta} }    \leq   \f{\Phi_y^h(t)}{t   \log(t)^{1+ \eps} }    ,

\]
as $t\to\infty$, uniformly among $h>0,y>g(h)$, as required for $\l(\ref{1'}\r)$.

 Now we prove   $(\mbox{\hyperref[SS*]{$S$}})\leq \Phi_y^h(t) u(t) \ov(g(t))  $.
The probability in $\l(\ref{S0}\r)$ converges to 0 as $t\to\infty$ by $\l(\ref{S2}\r)$, since $\lim_{t\to\infty}(*)=0$, and  this convergence is  uniform among $h>0,y>g(h)$ by Lemma \mbox{\ref{1.1}}, so for suitable $u$, $(S)\leq  \Phi_y^h(t) u(t) \ov(g(t))  $  as $t\to\infty$, uniformly  in $y,h$, as required for $\l(\ref{2'}\r)$.

 \begin{comment}
 \p{Proof for (\mbox{\hyperref[SS*]{$S$}}), Case (\hbox{\hyperref[case2]{ii}})}  To show $(\mbox{\hyperref[SS*]{$S$}})\leq \Phi_y^h(t) u(t) \ov(g(t))  $ in case (\hbox{\hyperref[case2]{ii}}), we  show the probability in $\l(\ref{S0}\r)$ converges to 0 as $t\to\infty$, uniformly.  By   Lemma \mbox{\ref{keylemma}} with $H(t)=1/\log(\log(t))$, as $t\to\infty$, uniformly in    $h>0,y>g(h)$ by~Lemma~\mbox{\ref{1.1}},
  \[
    \pp\l( X_{t}^{(0,   c g_{y}^{h}(t))}       >   \f{ g_{y}^{h}(t)     }{   2 \log(t)^\gamma  }           \r)      \overset{\ref{keylemma}}\leq   \exp\l( (*)   \r)   \f{1}{\log(\log(t))},
  \]
  \b{align*}
  (*)&\overset{\phantom{\ref{1.1}}}\lesssim      t \log\l(  \log(\log(t))   \r)   \log(\log(t))^{ \f{    c  }{  2\log(t)^\gamma  }   }  \ov\l(       c g_{y}^{h}(t)   \r)        \f{    c  }{  2\log(t)^\gamma  }
\\
  &\overset{\ref{1.1}}\lesssim  t  \log\l(  \log(\log(t))   \r)   \log(\log(t))^{ \f{    c  }{  2\log(t)^\gamma  }   }  \ov\l(   g(t)   \r)        \f{    c  }{  2\log(t)^\gamma  }.
  \end{align*}
  One can   verify    $  \lim_{t\to\infty}   \log\l(  \log(\log(t))   \r)   \log(\log(t))^{     c  /(  2\log(t)^\gamma  )   }     /( 2\log(t)^\gamma  )~\hspace{-4pt}=~\hspace{-4pt}0$, and then by $\l(\ref{cond2}\r)$, $ \lim_{t\to\infty}t \ov(g(t)) =  0$, and    it follows that   $\lim_{t\to\infty}(*)=0$.

\n     Hence  the probability in $\l(\ref{S0}\r)$ converges to 0 as $t\to\infty$, uniformly in $y,h$. So  $(\mbox{\hyperref[SS*]{$S$}})\leq  \Phi_y^h(t) u(t) \ov(g(t))  $ for suitable $u$, uniformly in $y,h$, as required~for~$\l(\ref{2'}\r)$.

 \end{comment}

\p{Partitioning  (\mbox{\hyperref[SS*]{$S^*$}})} Now we   partition $(\mbox{\hyperref[SS*]{$S^*$}})$. For $\beta$ as in $\l(\ref{condt}\r)$, $\gamma= (1-\alpha)/(2+\alpha)$, and $\d:=1+\gamma$, write $g_\d(t):= g(t)/\log(t)^\d$. Recall the notation $\l(\ref{D}\r)$. Then
\b{align}
\nonumber
(\mbox{\hyperref[SS*]{$S^*$}}) &=     \ov(g(t)) \int_0^t  \pp\l(  \oo_{s,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  \ ; \      X_{s-}^{(0,   c g_{y}^{h}(t))}    >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }   \r)   ds
\\
\nonumber
&=   \ov(g(t)) \int_0^t  \pp\l(  \oo_{s,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  \ ; \      X_{s-}^{(0,   c g_{y}^{h}(t))}    >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }     \ ; \    \D^{g_\d(t)} \leq s        \r)   ds
\\
\nonumber
&+  \ov(g(t)) \int_0^t  \pp\l(  \oo_{s,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  \ ; \      X_{s-}^{(0,   c g_{y}^{h}(t))}    >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }     \ ; \      \D^{g_\d(t)} >s     \r)   ds
\\
\label{S1*S2*}
&=: (S_1^*) + (S_2^*).
\end{align}

      \p{Proof for (\mbox{\hyperref[S1*S2*]{$S_1^*$}})}
       Disintegrating on the value of $ \D^{g_{\d}(t)} $, by $\l(\ref{s-}\r)$~and~$\l(\ref{phiyh}\r)$,
      \begin{align}
      \nonumber
      ( \mbox{\hyperref[S1*S2*]{$S_1^*$}})  &\overset{\phantom{\l(\ref{phiyh}\r)}}\leq      \ov( g(   t   )   )   \ov (    g_{\d}( \hspace{-0.5pt} t \hspace{-0.5pt} )   )  \hspace{-4pt}  \int_0^t \hspace{-4pt}  \int_0^s   \hspace{-4pt} \pp \Big( \hspace{-1.5pt} \oo_{ \hspace{-2.5pt} s,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  \hspace{-1pt} ; \hspace{-1pt}      X_{s-}^{(0,   c g_{y}^{h}(t))} \hspace{-2.5pt}   >  \hspace{-2.5pt}  \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }     \big|   \D^{g_{\d}(t)} \hspace{-2.5pt}=\hspace{-2pt} v       \hspace{-2pt} \Big) \hspace{-1pt}  dv  ds
  \\
  \nonumber
        &\overset{\phantom{\l(\ref{phiyh}\r)}}\leq    \ov( g(   t   )   )   \ov (    g_{\d}( \hspace{-0.5pt} t \hspace{-0.5pt} )   )  \hspace{-4pt}  \int_0^t \hspace{-4pt}  \int_0^t   \hspace{-4pt} \pp \Big( \hspace{-1.5pt} \oo_{ \hspace{-2.5pt} v,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  \hspace{-1pt} ; \hspace{-1pt}      X_{t}^{(0,   c g_{y}^{h}(t))} \hspace{-2.5pt}   >  \hspace{-2.5pt}  \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }     \big|   \D^{g_{\d}(t)} \hspace{-2.5pt}=\hspace{-2pt} v       \hspace{-2pt} \Big) \hspace{-1pt}  dv  ds
    \\

      &\overset{\hspace{-3pt}\l(\ref{s-}\r)}\leq     t     \ov( g(   t   )   )   \ov (    g_{\d}( \hspace{-0.5pt} t \hspace{-0.5pt} )   )    \int_0^t    \pp \Big(   \oo_{   v,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  \Big)    dv

    \label{s1*}

       \overset{\l(\ref{phiyh}\r)}\leq      t \ov(g(t))  \ov(g_{\d}(t))         \Phi_y^h(t).

      \intertext{As $\ov$ is \mbox{\hyperref[rv]{regularly varying}} at $\infty$ of index $\alpha$,   applying Potter's theorem  \cite[Theorem 1.5.6]{bgt89} to $\ov(g(t))$ and $\ov(g_\delta(t))$, for arbitrarily small $\tau>0$, as $t\to\infty$,}

      \nonumber
     ( \mbox{\hyperref[S1*S2*]{$S_1^*$}}) &\lesssim t   \ov(g_{\beta}(t))^2  \log(t)^{   -\alpha \beta    -\alpha(\beta-\delta)     +\beta\tau   + (\beta-\d)\tau }    \Phi_y^h(t),

          \intertext{$g_\beta(t)  :=   g(t)/\log(t)^\beta$,   $\beta$ as in $\l(\ref{condt}\r)$. By $\l(\ref{condt}\r)$, $\lim_{t\to\infty} t \ov(g_\beta(t))   =  0$,~so~as~$t  \to  \infty$,}

    \nonumber
    ( \mbox{\hyperref[S1*S2*]{$S_1^*$}})   &\leq   t^{-1}   \log(t)^{   -\alpha \beta    -\alpha(\beta-\delta)     +\beta\tau   + (\beta-\d)\tau }   \Phi_y^h(t).
\end{align}

\n  Now,   $\d=1+\gamma = 1+ (1-\alpha)/(2+\alpha)$, and  $-\alpha\beta - \alpha(\beta- \d )< -1$ by $\l(\ref{condt}\r)$. So taking   $\tau$ small enough, we  conclude that     $( \mbox{\hyperref[S1*S2*]{$S_1^*$}}) \lesssim   t^{-1}\log(t)^{-1-\eps}  \Phi_y^h(t) $ as $t\to\infty$, uniformly among $h>0,y>g(h)$, as required for $\l(\ref{1'}\r)$.

       \n To show   $( \mbox{\hyperref[S1*S2*]{$S_1^*$}}) \leq  \Phi_y^h(t) u(t) \ov(g(t))  $,  one can verify that for each $\alpha\in(0,1)$ and for $\beta$ as in $\l(\ref{condt}\r)$,      $\d =  1+ (1-\alpha)/(2+\alpha) <    (1+2\alpha)/(2\alpha+\alpha^2)   < \beta$. Thus   $\lim_{t\to\infty} t\ov(  g_\d(t))=0$ by $\l(\ref{condt}\r)$. Then by $\l(\ref{s1*}\r)$, for suitable $u$, uniformly in $y>g(h),h>0$ as $t  \to  \infty$,
\(
      ( \mbox{\hyperref[S1*S2*]{$S_1^*$}})        \lesssim   t \ov(g(t))  \ov(g_{\d}(t))         \Phi_y^h(t)   \leq    u(t) \ov(g(t)) \Phi_y^h(t)  $,  as required~for~$\l(\ref{2'}\r)$.

\begin{comment}

      \p{Proof for  (\mbox{\hyperref[S1*S2*]{$S_1^*$}}), Case (\hbox{\hyperref[case2]{ii}})} Noting that  $g_\d(t) \leq   g(t)  $ for all large   $t$,  as $t\to\infty$,
      \b{align*}
      ( \mbox{\hyperref[S1*S2*]{$S_1^*$}})   &=  \ov(g(t)) \int_0^t  \pp\l(  \oo_{s,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  \ ; \      X_{s-}^{(0,   c g_{y}^{h}(t))}    >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }     \ ; \    \D^{g_\d(t)} \leq s        \r)   ds
      \\
      &\leq          \ov(g(t)) \int_0^t  \pp\l(  \oo_{s,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  \ ; \      X_{s-}^{(0,   c g_{y}^{h}(t))}    >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }     \ ; \    \D^{g(t)} \leq s        \r)   ds.

         \intertext{Disintegrating on the value of $\D^{g(t)}$ and applying $\l(\ref{s-}\r)$,}

      ( \mbox{\hyperref[S1*S2*]{$S_1^*$}})     &\overset{\phantom{\l(\ref{s-}\r)}}\leq      \ov(  g(   t  ))^2    \int_0^t     \int_0^s   \pp \Big(   \oo_{   s,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  ;       X_{s-}^{(0,   c g_{y}^{h}(t))}    \hspace{-1pt} >     \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }     \big|     \D^{g(t)} =    v    \Big)        dv  ds
      \\
       &\overset{\phantom{\l(\ref{s-}\r)}}\leq      \ov(  g(   t  ))^2    \int_0^t     \int_0^s   \pp \Big(   \oo_{   v,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }     \big|     \D^{g(t)} =    v    \Big)        dv  ds
       \\
         &\overset{\l(\ref{s-}\r)}=     \ov(  g(   t  ))^2    \int_0^t     \int_0^s   \pp \Big(   \oo_{   v,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }         \Big)        dv  ds
     \\
    &\overset{\phantom{\l(\ref{s-}\r)}}\leq t        \ov(  g(   t  ))^2    \int_0^t  \pp \big(   \oo_{   v }^{g_y^h}      \big)      dv

         \\
          &\overset{\phantom{\l(\ref{s-}\r)}}\lesssim    t \ov(g(t))^2  \Phi_y^h(t) .
         \end{align*}
     Now, $ \lim_{t\to\infty} t \ov(g(t))\hspace{-1.5pt}=\hspace{-1.5pt}0$ by $\l(\ref{cond2}\r)$, and hence  $( \mbox{\hyperref[S1*S2*]{$S_1^*$}})\hspace{-1.5pt}\leq \Phi_y^h(t)  u(t) \ov(g(t))  $~for~{suitable}~$u$, uniformly among $h>0,y>g(h)$ as $t\to\infty$, as required for $\l(\ref{2'}\r)$.
      \end{comment}

  \p{Proof for (\mbox{\hyperref[S1*S2*]{$S_2^*$}})}
  Note for $g_\d(t):=g(t)/\log(t)^\d$,  by Lemma \mbox{\ref{1.1}},  for all $h>0$, $y>g(h)$, and for all large enough $t$, $g_\d(t) \leq c g_y^h(t)$, so as $t\to\infty$,
  \b{align}
  \nonumber
  (\mbox{\hyperref[S1*S2*]{$S_2^*$}})  &=      \ov(g(t)) \int_0^t  \pp\l(  \oo_{s,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }  \ ; \      X_{s-}^{(0,   c g_{y}^{h}(t))}    >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }     \ ; \      \D^{g_{\d}(t)} >s     \r)   ds
\\
\nonumber
 &=      \ov(g(t)) \int_0^t  \pp\l(  \oo_{s,X^{(0,  g_{\d}(t))} }^{ g_{y}^{h} }  \ ; \      X_{s-}^{(0,  g_{\d}(t))}    >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }     \ ; \      \D^{g_{\d}(t)} >s     \r)   ds
 \\
\nonumber
 &\leq     \ov(g(t)) \int_0^t  \pp\l(  X_{s-}^{(0,  g_{\d}(t))}    >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }        \r)   ds

  \leq  t \ov(g(t))   \pp\l(     X_{t}^{(0,  g_{\d}(t))}    >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }          \r)   .

  \intertext{For  $g_\beta(t):=g(t)/\log(t)^\beta$, with $\beta$ as in $\l(\ref{condt}\r)$,   applying Potter's theorem \cite[Theorem 1.5.6]{bgt89} to $\ov(g(t))$,  for arbitrarily small $\tau>0$, as $t\to\infty$,  by Lemma \mbox{\ref{1.1}},}

\nonumber
      (\mbox{\hyperref[S1*S2*]{$S_2^*$}})  &\overset{\phantom{\ref{1.1}}}\lesssim     t \ov(g_\beta(t))   \log(t)^{  -\alpha\beta + \tau \beta }        \pp\l(     X_{t}^{(0,  g_{\d}(t))}    >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }          \r)
     \\
      \nonumber
      &\overset{\ref{1.1}}\leq       t \ov(g_\beta(t))   \log(t)^{  -\alpha\beta + \tau \beta }        \pp\l(     X_{t}^{(0,  g_{\d}(t))}    >   \f{(1-A^{-1}) g (t)          }{   2 \log(t)^\gamma  }          \r)    .

  \intertext{Applying Lemma \mbox{\ref{keylemma}} with   $H(t) = 1/(t  \log(t)^{1 + \eps   -\alpha\beta    })$, $\eps>\tau\beta $,   then by $\l(\ref{condt}\r)$ and Lemma \mbox{\ref{1fyh}}, uniformly in $h>0,y>g(h)$ by Lemma \mbox{\ref{1.1}}, as $t\to\infty$,  }

  \nonumber
   (\mbox{\hyperref[S1*S2*]{$S_2^*$}})  &\overset{\hspace{1.5pt}\ref{keylemma}  \hspace{1.5pt}  }\lesssim           t \ov(g_\beta(t))   \log(t)^{  -\alpha\beta + \tau \beta }  \exp ((*)) \f{1}{t  \log(t)^{1 + \eps   -\alpha\beta    }}

   \overset{ \hspace{1.5pt} \l(\ref{condt}\r)  \hspace{1.5pt}   }=  o(1) \times  \exp ((*)) \f{1}{t  \log(t)^{1 + \eps   -\tau\beta    }}
   \\
   \label{S2*bit}
   &\overset{\ref{1fyh}}\leq      o(1) \times       \exp ((*)) \f{1}{t  \log(t)^{1 + \eps   -\tau\beta    }}    \f{    \Phi_y^h(t) }{    f(y)-h},
   \\
   \nonumber
    (*) &\overset{\phantom{\ref{1fyh}}}\lesssim   t  \log\l(t    \log(t)^{1+\eps -\alpha\beta}    \r)    \l(t    \log(t)^{1+\eps -\alpha\beta}    \r)^{   \f{2  \log(t)^{\gamma-\d}}{1-A^{-1}}  }   \ov(g_\d(t))   \log(t)^{\gamma - \d   }.

  \intertext{Now, if $\lim_{t\to\infty} (*)     =0$, then $(\mbox{\hyperref[S1*S2*]{$S_2^*$}})     \lesssim      t^{-1}        \log(t)^{-1-\eps}           \Phi_y^h(t)  /       (   f( y)         -      h) $. Indeed,
    }

 \nonumber
   (*) &\lesssim   t  \log\l(t    \log(t)^{1+\eps -\alpha\beta}    \r)    \l(t    \log(t)^{1+\eps -\alpha\beta}    \r)^{  \f{2  \log(t)^{\gamma-\d} }{ 1- A^{-1} }   }   \ov(g_\d(t))   \log(t)^{\gamma - \d   }
\\
\nonumber
  &=    t  \ov(g_\d(t))      t^{ \f{ 2\log(t)^{\gamma-\d}  }{ 1- A^{-1} }     }       \log\big(t    \log(t)^{1+\eps -\alpha\beta}    \big)     \log(t)^{  \l( 1+\eps -\alpha\beta   \r) \f{2\log(t)^{\gamma-\d}}{ 1- A^{-1} }    +\gamma-\d   }.

   \intertext{Now, $\d  >  \gamma$, so  $\lim_{t\to\infty}\log(t)^{\gamma-\d}    =  0$, and for arbitrarily small $\kappa  >  0$, as~$t  \to  \infty$,
   }
   \nonumber
    (*) &\lesssim  t  \ov(g_\d(t))     t^{ \f{ 2\log(t)^{\gamma-\d}  }{ 1- A^{-1} }     }         \log\big(t    \log(t)^{1+\eps -\alpha\beta}    \big)     \log(t)^{  \l( 1+\eps -\alpha\beta   \r) \kappa   +\gamma-\d   }.

\intertext{As $t\to\infty$, $\log(t\log(t)^{1+\eps - \alpha\beta})\lesssim\log(t)$. Applying Potter's theorem    \cite[Theorem 1.5.6]{bgt89}
   to $  \ov(g_\d(t)) $, for $\beta$ as in $\l(\ref{condt}\r)$ and arbitrarily small $c  >  0$,~as~$t  \to  \infty$,  }

\nonumber
 (*) &\lesssim     t  \ov(g_\beta(t))         t^{ \f{ 2\log(t)^{\gamma-\d}  }{ 1- A^{-1} }     }         \log(t)^{ 1 +  \l( 1+\eps -\alpha\beta   \r)\kappa -(\beta-\d)\alpha   + (\beta-\delta)c  +\gamma-\d    }.

  \intertext{Recalling     $\gamma   -  \d     =     -   1$,  \(

  t^{ \f{ 2\log(t)^{\gamma-\d}  }{ 1- A^{-1} }     }      =      e^{  \f{  2\log(t)^{1+\gamma-\d}  }{ 1- A^{-1}  }   }   \hspace{-1pt}= \hspace{-1pt} e^{\f{2}{1-A^{-1}}}.

\) Then as  $\lim_{t\to\infty} t \ov(g_\beta(t)) = 0  $ by $\l(\ref{condt}\r)$,  using that $1+\gamma-\d=0$, as $t\to\infty$,}

  (*) &\lesssim       \log(t)^{ 1 +  \l( 1+\eps -\alpha\beta   \r)\kappa -(\beta-\d)\alpha   + (\beta-\delta)c  +\gamma-\d    }

    \label{777}

  =  \log(t)^{    \l( 1+\eps -\alpha\beta   \r)\kappa   -(\beta-\d)\alpha   + (\beta-\delta)c       }.

\end{align}
 Now, $\d<\beta$, so  $-(\beta-\d)\alpha<0$. Choosing $\kappa$, $c$ small enough that the   exponent in $\l(\ref{777}\r)$  is negative,  $\lim_{t\to\infty}(*)=0$. Then by $\l(\ref{S2*bit}\r)$,   uniformly in $h>0,y>g(h)$  as $t\to\infty$, $(\mbox{\hyperref[S1*S2*]{$S_2^*$}})\lesssim   t^{-1} \log(t)^{-1-\eps}  \Phi_y^h(t)/(f(y)-h) $,  as required for $\l(\ref{1'}\r)$.

  To prove $(\mbox{\hyperref[S1*S2*]{$S_2^*$}})\leq  \Phi_y^h(t)  u(t) \ov(g(t))  / (f(y)-h)  $,
    applying  Lemma \mbox{\ref{keylemma}} with $H(t)= 1/ (t \log(\log(t)))   $ to $\l(\ref{S2*bit}\r)$, as $t\to\infty$, uniformly in $h>0,y>g(h)$,
  \b{align*}
  (\mbox{\hyperref[S1*S2*]{$S_2^*$}})  &\overset{\l(\ref{S2*bit}\r)}\lesssim  t \ov(g(t))    \pp\l(     X_{t}^{(0,  g_{\d}(t))}    >   \f{  (1-A^{-1}) g(t)          }{   2 \log(t)^\gamma  }          \r)

  \overset{\hspace{1.5pt}\ref{keylemma}\hspace{1.5pt}}\leq        \f{\ov(g(t))}{\log(\log(t))}   \exp \l( (*)      \r)      ,

\\
  (*)  &\lesssim       t \log\l(    t \log(\log(t)) \r)      \l(   t \log(\log(t))  \r)^{      \f{ 2\log(t)^{\gamma-\d}   }{   (1-A^{-1})}     }    \ov(g_\d(t))      \log(t)^{\gamma-\d}        .

  \intertext{Recall $1+\gamma- \d=0$. As $    t   \log(\log(t))   \lesssim   t^2   $  for all large enough $t$, as $t\to\infty$,}

 (*)
  &\lesssim      t      \log(t)^{1+\gamma-\d}     t^{   \f{   4   \log(t)^{\gamma-\d}    }{ 1-A^{-1}}   }      \ov(g_\d(t))

    =       t       e^{  \f{   4 (1-A^{-1})  \log(t)^{1+\gamma-\d}   }{ 1-A^{-1}}   }      \ov(g_\d(t))

    =      t        e^{  \f{   4 (1-A^{-1})     }{ 1-A^{-1}}   }      \ov(g_\d(t))        .

  \end{align*}
  Now, since $ \d< \beta$, by $\l(\ref{condt}\r)$,  $   \lim_{t\to\infty}t\ov(g_\d(t)) = 0$, and      hence  $\lim_{t\to\infty}(*)=0$, and   we   conclude that   for suitable $u$,  $(\mbox{\hyperref[S1*S2*]{$S_2^*$}})\leq   \Phi_y^h(t) u(t) \ov(g(t)) / (f(y)-h)  $,  uniformly among $h>0,y>g(h)$ as $t\to\infty$, as required for $\l(\ref{2'}\r)$.

\begin{comment}

 \p{Proof for (\mbox{\hyperref[S1*S2*]{$S_2^*$}}), Case (\hbox{\hyperref[case2]{ii}})}  Recalling the notation $ \l(\ref{D}\r)$, $\l(\ref{ogyh}\r)$,  and the definition $\l(\ref{truncoo}\r)$,
 \b{align*}
  (\mbox{\hyperref[S1*S2*]{$S_2^*$}})  &=     \ov(g(t)) \int_0^t  \pp\l(  \oo_{s,X^{(0,   c g_{y}^{h}(t))} }^{ g_{y}^{h} }   ;       X_{s-}^{(0,   c g_{y}^{h}(t))}    >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }      ;       \D^{\f{ 1}{2}  c g_{y}^{h}(t)  } >s     \r)   ds
\\
  &\leq   t \ov(g(t))   \pp\l(     X_{t}^{(0,   \f{ 1}{2}  c g_{y}^{h}(t)  )}         >   \f{ g_{y}^{h}(t)          }{   2 \log(t)^\gamma  }          \r)   .

\intertext{By  Lemma \mbox{\ref{keylemma}} with $H(t)    =    1/(t \log(\log(t)))$, uniformly in $y,h$~by~Lemma~\mbox{\ref{1.1}},  }

  (\mbox{\hyperref[S1*S2*]{$S_2^*$}})     &\lesssim      \f{\ov(g(t))}{  \log(\log(t))   }    \exp\l(   (*) \r),
\\
    (*)  &\lesssim          t \log( t\log(\log(t)))   \    \l( t \log(\log(t))\r) ^{     \f{   c}{  4 \log(t)^\gamma }   }   \ov\l(      \f{1}{2}   c g_{y}^{h}(t)\r)           \f{   c}{  4 \log(t)^\gamma } .

      \intertext{For each arbitrarily small $\eta>0$,  as $t\to\infty$, $\log( t\log(\log(t))) \hspace{-2pt}  \leq \hspace{-2pt} t^{\eta}$ and   $ \l( t \log(\log(t))\r) ^{        c/(  4 \log(t)^\gamma )   }~\hspace{-7pt}\leq~\hspace{-7pt}~t^{\eta}$.  As $\ov$ is \mbox{\hyperref[rv]{regularly varying}} at $\infty$, by Lemma \mbox{\ref{1.1}},}

        (*)&\lesssim t^{1+2\eta}   \ov\l(      \f{   c g_{y}^{h}(t)}{2} \r)   \f{1}{   \log(t)^\gamma }
        \\
           &\lesssim t^{1+2\eta}   \ov\l(    g(t) \r)      .
        \end{align*}
Now, $\lim_{t\to\infty} t^{1+\beta} \ov(g(t))=0$ by $\l(\ref{cond2}\r)$,  so taking $\eta <  \beta/2$,   by Lemma \mbox{\ref{1fyh}}, $(\mbox{\hyperref[S1*S2*]{$S_2^*$}})\leq   \Phi_y^h(t)  u(t) \ov(g(t)) / (f(y)-h)  $ for suitable $u$ as $t\to\infty$, uniformly in $h  >  0,y  >  g(h)$, as required for $\l(\ref{2'}\r)$, and     the proof of Lemma \mbox{\ref{lemma2}}~is~complete.\end{comment}

 \end{proof}

\begin{comment}

\p{Optimising Choices of $\beta,\gamma,\d$ in case (\hbox{\hyperref[case1]{i}})}   \textcolor{green}{For (1), we need:}
\b{equation}
\gamma+\alpha\beta >1;
\e{equation}
\b{equation}
\alpha\beta>\f{1}{2};
\e{equation}
\b{equation}
\gamma,\beta,\d>0;
\e{equation}
\b{equation}
2\alpha \beta    -\alpha\d >1;
\e{equation}
\b{equation}
1+ \gamma\leq\d .
\e{equation}
 \textcolor{red}{For (2), we need:}
\b{equation}
\gamma,\beta,\d>0;
\e{equation}
\b{equation}
 \d\leq\beta   ;
\e{equation}
\b{equation}
 1+\gamma\leq \d.
\e{equation}
The goal here is to minimise the value of $\beta$ by varying $\d$ and $\gamma$.    We want $\beta$ to be universal, but $\delta$ and $\gamma$ can be different for $\l(\ref{1}\r)$ and $\l(\ref{2}\r)$ if needed.    Let us consider the choices for $\l(\ref{1}\r)$ first.

\n First consider $\d$. It is clear from (9) and (12) that we need to minimise $\d$ in order to minimise $\beta$, so it is optimal to take  $\d= 1+\gamma$ so (10) is satisfied.

\n Now consider $\gamma$. From (6) and (9),  the range of permissible values are given by $    1-\alpha\beta < \gamma < \f{2\alpha\beta -1}{  \alpha }= 2\beta - \f{1}{\alpha}$.

\n  Then from (6),(7), and (9),     \[
\beta> \f{ 1 - \gamma}{\alpha}   \vee    \f{1}{2\alpha}   \vee  \f{1 + \alpha(1+\gamma)}{  2\alpha  }              =    \f{ 1 - \gamma}{\alpha}   \vee   \l( \f{1  }{  2\alpha  }     + \f{1+\gamma}{2}   \r).
\]
Then it is simply a matter of solving to maximise this, which happens when $ \f{ 1 - \gamma}{\alpha}   =   \l( \f{1  }{  2\alpha  }     + \f{1+\gamma}{2}   \r)$, that is
\[
    \f{1}{\alpha}   - \f{1}{2\alpha}   - \f{1}{2}  =    \l(    \f{1}{2}   + \f{1}{\alpha} \r)   \gamma
\]
 \[ \iff    \gamma=    \f{        \f{1}{2\alpha}   - \f{1}{2}    }{   \f{1}{2}   + \f{1}{\alpha}    }    =    \f{    1 - \alpha   }{   \alpha + 2    }.
 \]
Then  for arbitrarily small $\eps>0$,
 \[
\beta =    \f{ 1-   \l(     \f{    1 - \alpha   }{   \alpha + 2    } \r)  }{\alpha}    +\eps  =       \f{1+2\alpha   }{ \alpha(2+\alpha)  }   +\eps .
\]
Note that when $\alpha=1/2$, this gives $\beta = 8/5+\eps$, as in \ks, and such a choice of $\beta,\gamma,\d$ can easily be seen as consistent with (12), so with this choice of $\beta$, the proof is complete.
  \end{comment}

\section{Proof of Lemma  \hbox{\ref{l6}}} \label{l6proof}

  \b{proof}[Proof of Lemma \mbox{\ref{l6}}]

    Let $T_{g(h)}$ be the time when $X$   passes above $g(h)$, and let $S_{\Delta_1^{g(h)}}
    $ be the size of $X$'s first jump larger than $g(h)$. For each~$y  >  K$,
  \b{align}
  \nonumber
      \pp\l(   X_h \in g(h) dy ;   \cal{O}_h    \r)

 &=         \pp\l(   X_h \in g(h) dy ;      X_{T_{g(h)}} \leq  \f{g(h)y}{2}      ;  \cal{O}_h    \r)
\\
\nonumber
 &+      \pp\l(   X_h \in g(h) dy ;      X_{T_{g(h)}} >  \f{g(h)y}{2}     ;   S_{\Delta_1^{g(h)}}  < \f{g(h)y}{2}      ;\cal{O}_h    \r)
 \\
 \nonumber
 &+      \pp\l(   X_h \in g(h) dy ;      X_{T_{g(h)}} >  \f{g(h)y}{2}      ;  S_{\Delta_1^{g(h)}}  \geq \f{g(h)y}{2}      ; \cal{O}_h    \r)
 \\
 \label{starting}
 &=:   \sigma_h^1(dy)   +  \sigma_h^2(dy)    + \sigma_h^3(dy).
   \end{align}
We     will bound $ \pp\l(   X_h \in g(h) dy ;   \cal{O}_h    \r)   $ by bounding  these 3 terms separately.

  \p{Upper Bound for \mbox{\hyperref[starting]{$\sigma_h^1(dy)$}}} Disintegrating on the values of $T_{g(h)}$ and $X_{T_{g(h)}}$,  then as $\cal{O}_h \cap \{  \D^{g(h)} \in ds   \}   =  \cal{O}_{ \D^{g(h)} - }  \cap \{  \D^{g(h)} \in ds   \}$,  applying $\l(\ref{s-}\r)$ and the Markov property, with $\pp(X_t\in dx)=f_t(x)dx  $,
  \b{align}
\nonumber
  \mbox{\hyperref[starting]{$\sigma_h^1(dy)$}} &=   \int_{s=0}^{h} \int_{w=1}^{\f{y}{2}}        \mathbb{P}\l(   X_h  \in    g(h)  dy ;   X_{T_{g(h)}}    \in g(h) dw ;   T_{g(h)} \in ds  ;  \cal{O}_h  \r)
\\
\nonumber
    &=  \int_{0}^{h} \int_{1}^{\f{y}{2}}       f_{h-s} (g(h)  (y-w))g(h)  dy \pp\l(    X_{T_{g(h)}}    \in g(h) dw ;   T_{g(h)}  \in ds   ;  \cal{O}_h      \r).
    \end{align}

Now, $g(h)  (y-w) > g(h)y/2>g(h)+x_0\geq  g(h-s) + x_0$, for all large enough $h$, with   $x_0$ as  in Definition \mbox{\ref{case1a}}, so   $\l(\ref{l6cond}\r)$ applies to $  f_{h-s} (g(h)  (y-w))$.   Applying $\l(\ref{2.10.3}\r)$, since $ y-w\geq y/2$ and $L$ is \mbox{\hyperref[sv]{slowly varying}} at $\infty$, uniformly in $y>K$ by  \cite[Theorem 1.2.1]{bgt89},  as $h\to\infty$,

\begin{align}
  \nonumber
     \mbox{\hyperref[starting]{$\sigma_h^1(dy)$}}   \hspace{-4pt}   &\overset{(\ref{l6cond})}\lesssim  \hspace{-4pt}  \int_{0}^{h} \hspace{-4pt} \int_{1}^{\f{y}{2}}   (h  -  s) u(g(h)(y  -  w)) g(h) dy \pp  \l(    X_{T_{g(h)}}    \in g(h) dw ;    T_{g(h)} \in ds;  \cal{O}_h    \r)
\\
 \nonumber
    &\overset{(\ref{2.10.3})}\lesssim \hspace{-4pt} \int_{0}^{h} \hspace{-4pt} \int_{1}^{\f{y}{2}}  \f{(h  -  s)}{ g(h)^{\alpha} }  (y  -  w)^{-1-\alpha} L(g(h)(y-w)) dy

  \pp  \l( \hspace{-2pt}   X_{T_{g(h)}}  \hspace{-4pt}   \in g(h) dw ;    T_{g(h)} \in ds;  \cal{O}_h \hspace{-2pt}   \r)
  \\
  \nonumber
   &\overset{\phantom{(\ref{l6cond})}}\lesssim \hspace{-4pt} \int_{0}^{h}   \hspace{-4pt}   \int_{1}^{\f{y}{2}}  \f{h}{ g(h)^{\alpha}}\f{L(g(h))}{L(g(h))} y^{-1-\alpha} L\l(\f{g(h)y}{2}\r) dy

  \pp  \l(    X_{T_{g(h)}}    \in g(h) dw ;   T_{g(h)} \in ds  ;  \cal{O}_h  \r)
\\
  \nonumber
  &\overset{\phantom{(\ref{l6cond})}}\lesssim   y^{-1-\alpha}\f{ L\l(g(h)y\r)}{L(g(h))} dy   \f{h L(g(h)) }{ g(h)^{\alpha}}   \hspace{-1pt}  \int_{0}^{h} \hspace{-1pt} \int_{1}^{\f{y}{2}}      \hspace{-1pt}
 \pp  \l(    X_{T_{g(h)}}  \hspace{-1pt}  \in \hspace{-1pt} g(h) dw ;   T_{g(h)} \hspace{-1pt} \in   \hspace{-1pt} ds ;  \cal{O}_h   \r)
\\

  &\overset{\phantom{(\ref{l6cond})}}\leq    y^{-1-\alpha}\f{ L\l(g(h)y\r)}{L(g(h))} dy \    h   \ov(g(h)) \
 \pp\l(  \cal{O}_h    \r)

 \label{sigma1bound}

 \overset{(\ref{condt})}=   o(1) \times    y^{-1-\alpha}\f{ L\l(g(h)y\r)}{L(g(h))} \pp(\cal{O}_h) dy    ,
\end{align}
   where, recalling   $ g(h)^{-\alpha} L(g(h))=   \ov(g(h)) $,   the last inequality follows by~$\l(\ref{condt}\r)$.

  \p{Simplifying the Expressions for \mbox{\hyperref[starting]{$\sigma_h^2(dy)$}} and \mbox{\hyperref[starting]{$\sigma_h^3(dy)$}}}
     Recall     $\l(\ref{D}\r)$, and      $\mbox{\hyperref[starting]{$\sigma_h^2(dy)$}} + \mbox{\hyperref[starting]{$\sigma_h^3(dy)$}}    = \pp (
   X_h \in g(h) dy ;      X_{T_{g(h)}}   >    g(h)y/2    ;  \cal{O}_h    )$.
  Note   $g(h)y/2   >   2  g(h) $  for  $y  >  K$.

  As $T_{g(h)}$ is the first passage time above  $g(h)$,   if $X_{T_{g(h)}}   >   2g(h)$, then  $X$   crosses $g(h)$ by a jump larger than $g(h)$, so

       since $T_{g(h)}  \leq  \Delta_1^{g(h)}$, $T_{g(h)}  =   \Delta_1^{g(h)}$.

  \n  Then since $X_t<g(h)$ for all $t <  T_{g(h)}$, it follows that $X_{T_{g(h)} -} = X_{\Delta_1^{g(h)}  -}   < g(h)$, as $X$   has c\`adl\`ag sample paths, almost surely.

  Moreover, if $\cal{O}_h$ holds, then $X$   crosses $g(h)$ by time $h$, so  $\Delta_1^{g(h)} = T_{g(h)}   \leq h$.  Thus:

     \begin{equation}
     \label{firstinclusion}
   \l\{   \hspace{-2.5pt}   X_{T_{g(h)}} \hspace{-2.5pt} > \hspace{-2.5pt} \f{g(h)y}{2}     ;   S_{\Delta_1^{g(h)}}  \hspace{-2.5pt}<\hspace{-2.5pt} \f{g(h)y}{2}      ;\cal{O}_h     \r\}   \hspace{-2.5pt}      \subseteq       \hspace{-2.5pt}       \l\{     \Delta_1^{g(h)}     \leq    h ;       X_{\Delta_1^{g(h)}  -}  \hspace{-3.5pt} < g(h)        ;   S_{\Delta_1^{g(h)}} \hspace{-2.5pt} <  \hspace{-2.5pt}  \f{g(h)y}{2}      ;\cal{O}_h       \r\}    ,
  \end{equation}
  and therefore  we can bound $  \sigma_h^2(dy) $ by
  \b{align}
  \nonumber
  \mbox{\hyperref[starting]{$\sigma_h^2(dy)$}}   &=      \pp\l(   X_h \in g(h) dy ;      X_{T_{g(h)}} >  \f{g(h)y}{2}     ;   S_{\Delta_1^{g(h)}}  < \f{g(h)y}{2}      ;\cal{O}_h    \r)
\\
\label{sigma2}
  &\leq         \pp \l(  \hspace{-2pt} X_h \in g(h) dy ;    \Delta_1^{g(h)}   \leq h ;       X_{\Delta_1^{g(h)}  -}  \hspace{-2pt} <   \hspace{-2pt}  g(h)        ;   S_{\Delta_1^{g(h)}} \hspace{-2pt} < \hspace{-2pt} \f{g(h)y}{2}      ;\cal{O}_h \hspace{-2pt}   \r)  \hspace{-2pt}  .

  \end{align}
 For \mbox{\hyperref[starting]{$\sigma_h^3(dy)$}},   the converse inclusion to $\l(\ref{firstinclusion}\r)$ holds too, that is, if   $    \Delta_1^{g(h)}   \leq h$,         $  X_{\Delta_1^{g(h)}  -}  \hspace{-0.15cm}    <   \hspace{-0.05cm}   g(h)   $, $S_{\Delta_1^{g(h)}}  \geq g(h)y/2$, and       $ \cal{O}_h $ hold, then we have

       \[
   X_{T_{g(h)}} =   X_{\Delta_1^{g(h)}}  \geq   X_{\Delta_1^{g(h)}}  -   X_{\Delta_1^{g(h)}-}   = S_{\Delta_1^{g(h)}} >   g(h)y/2  ,
   \]
  and therefore   $\sigma_h^3(dy)  $ satisfies
     \b{align}
    \nonumber
    \mbox{\hyperref[starting]{$\sigma_h^3(dy)$}}    &=      \pp\l(   X_h \in g(h) dy ;      X_{T_{g(h)}} >  \f{g(h)y}{2}      ;  S_{\Delta_1^{g(h)}}  \geq \f{g(h)y}{2}      ; \cal{O}_h    \r)
  \\
    \label{contain}
    &=        \pp    \l(\hspace{-0.1cm}    X_h \in g(h) dy ;       \Delta_1^{g(h)}   \leq h ;       X_{\Delta_1^{g(h)}  -}  \hspace{-0.15cm}    <   \hspace{-0.05cm}   g(h)        ;  S_{\Delta_1^{g(h)}}  \hspace{-0.05cm} \geq \hspace{-0.05cm} \f{g(h)y}{2}      ; \cal{O}_h  \hspace{-0.075cm}  \r) .
  \end{align}

    \p{Upper Bound for \mbox{\hyperref[starting]{$\sigma_h^2(dy)$}}}
     By $\l(\ref{sigma2}\r)$, disintegrating on the values of $\Delta_1^{g(h)}  $, $ X_{\Delta_1^{g(h)}-}  $, and   $ S_{\Delta_1^{g(h)}}   $,
       by the Markov property, with $\pp(X_t\in dx)=f_t(x)dx  $,
  \b{align}
  \nonumber
  \mbox{\hyperref[starting]{$\sigma_h^2(dy)$}} &\overset{\l(\ref{sigma2}\r)}\leq       \pp\l(   X_h \in g(h) dy ;    \Delta_1^{g(h)}   \leq h ;       X_{\Delta_1^{g(h)}  -}   < g(h)        ;   S_{\Delta_1^{g(h)}}  < \f{g(h)y}{2}      ;\cal{O}_h    \r)
\\
\nonumber
  &\overset{\phantom{\l(\ref{l6cond}\r)}}=   \int_{s=0}^{h}     \int_{w=0}^{1}    \int_{    v=0}^{\f{y}{2}}          \pp\big(   X_h \in g(h) dy ;    \Delta_1^{g(h)}   \in ds ;       X_{\Delta_1^{g(h)}  -}   \in g(h) dw         ;
  \\
  \nonumber
  &\ \hspace{94pt}  S_{\Delta_1^{g(h)}}  \in g(h) dv    ;    \cal{O}_h    \big)
 \\
 \nonumber
  &\overset{\phantom{\l(\ref{l6cond}\r)}}= \int_{0}^{h}     \int_{0}^{1}    \int_0^{\f{y}{2}}      f_{h-s} ( g(h) (y-w-v) ) g(h)   dy
  \\
  \nonumber   &\    \hspace{2cm}  \times  \pp\l(    \Delta_1^{g(h)}   \in ds ;       X_{\Delta_1^{g(h)}  -}   \in g(h) dw         ;   S_{\Delta_1^{g(h)}}  \in g(h) dv    ;\cal{O}_h    \r).
  \end{align}

Note    $y-w-v > y/3>K/3$ for $w\leq1$,  $ v\leq y/2$.  So  as $h\to\infty$, $g(h)(y-w-v) \geq g(h-s) + x_0$,    so    we can apply  $\l(\ref{l6cond}\r)$, and we apply $\l(\ref{2.10.3}\r)$ too. Now, $g(h)^{-\alpha} L(g(h))  =  \ov(g(h))$  for   $L$   \mbox{\hyperref[sv]{slowly varying}} at $\infty$,  so by $\l(\ref{condt}\r)$,  uniformly in $y  >  K$   by the uniform convergence theorem \hspace{-6pt} \cite[Theorem 1.2.1]{bgt89},~as~$h  \to  \infty$,

\begin{align}
\nonumber
  \mbox{\hyperref[starting]{$\sigma_h^2(dy)$}}  &\overset{\l(\ref{l6cond}\r)}\lesssim \int_{0}^{h}     \int_{0}^{1}    \int_0^{\f{y}{2}}    (h-s) u (g(h)(y-w-v))  g(h) dy
  \\
   \nonumber
  &\  \hspace{2cm}    \times  \pp\l(    \Delta_1^{g(h)}   \in ds ;       X_{\Delta_1^{g(h)}  -}   \in g(h) dw         ;   S_{\Delta_1^{g(h)}}  \in g(h) dv    ;\cal{O}_h    \r)

\\
    \nonumber
       &\overset{\l(\ref{2.10.3}\r)}\lesssim   \int_{0}^{h}     \int_{0}^{1}    \int_0^{\f{y}{2}}     \f{   (h-s)      L(g(h)(y-w-v))      }{g(h)^{\alpha}  (y-w-v)^{1+\alpha}}   dy
  \\      \nonumber
    &\  \hspace{2cm}    \times  \pp\l(    \Delta_1^{g(h)}   \in ds ;       X_{\Delta_1^{g(h)}  -}   \in g(h) dw         ;   S_{\Delta_1^{g(h)}}  \in g(h) dv    ;\cal{O}_h    \r)

  \\
    \nonumber
       &\overset{\phantom{\l(\ref{2.10.3}\r)}}\lesssim   \int_{0}^{h}     \int_{0}^{1}    \int_0^{\f{y}{2}}  \f{   (h-s) }{g(h)^{\alpha}  y^{1+\alpha}}  L(g(h)y) dy
  \\      \nonumber
    &\  \hspace{2cm}    \times  \pp\l(    \Delta_1^{g(h)}   \in ds ;       X_{\Delta_1^{g(h)}  -}   \in g(h) dw         ;   S_{\Delta_1^{g(h)}}  \in g(h) dv    ;\cal{O}_h    \r)

     \\       \nonumber
     &\overset{\phantom{\l(\ref{l6cond}\r)}}\leq     \f{   h   L(g(h)) }{g(h)^{\alpha}  }   \int_{0}^{h}     \int_{0}^{1}    \int_0^{\f{y}{2}}  y^{-1-\alpha}    \f{L(g(h)y) }{L(g(h)} dy
  \\      \nonumber
  &\  \hspace{2cm}    \times  \pp\l(    \Delta_1^{g(h)}   \in ds ;       X_{\Delta_1^{g(h)}  -}   \in g(h) dw         ;   S_{\Delta_1^{g(h)}}  \in g(h) dv    ;\cal{O}_h    \r)

 \\
         \nonumber
  &\hspace{2pt}  \overset{\l(\ref{condt}\r)}=    o(1) \times    \int_{0}^{h}     \int_{0}^{1}    \int_0^{\f{y}{2}}       y^{-1-\alpha}    \f{L(g(h)y) }{L(g(h)} dy
  \\   \nonumber
   &\  \hspace{2cm}    \times  \pp\l(    \Delta_1^{g(h)}   \in ds ;       X_{\Delta_1^{g(h)}  -}   \in g(h) dw         ;   S_{\Delta_1^{g(h)}}  \in g(h) dv    ;\cal{O}_h    \r)
   \\

\label{sigma2bound}
  &\overset{\phantom{\l(\ref{l6cond}\r)}}\leq       o(1) \times  y^{-1-\alpha}  \f{L(g(h)y)}{L(g(h))} \pp(\cal{O}_h)  dy.
  \end{align}

  \begin{comment}

\n Now we consider \mbox{\hyperref[starting]{$\sigma_h^3(dy)$}}.      For $ \kappa=\kappa_h:=    2 +  x_0 /g(h) $, with $x_0$ as  in Definition~\mbox{\ref{case1a}},
      \b{align}
      \nonumber
      \mbox{\hyperref[starting]{$\sigma_h^3(dy)$}} &=       \pp\l(   X_h \in g(h) dy ;    \Delta_1^{g(h)}   \leq h ;       X_{\Delta_1^{g(h)}  -}   < g(h)        ;  S_{\Delta_1^{g(h)}}  \geq \f{g(h)y}{2}      ; \cal{O}_h    \r)
  \\
  \nonumber
  &=   \hspace{-3pt}  \pp  \hspace{-2pt}  \l(    \hspace{-3.5pt}   X_h  \hspace{-0.1cm}   \in  \hspace{-0.05cm}    g(h) dy ;   \hspace{-0.05cm}      \Delta_1^{g(h)}   \hspace{-3.5pt}   \leq   \hspace{-0.05cm}   h ;   \hspace{-1pt}    X_{        \Delta_1^{g(h)}  \hspace{-1pt} -}   \hspace{-4.5pt}  <  \hspace{-0.05cm} g(h)       ;   \hspace{-1pt}   S_{  \Delta_1^{g(h)}}       \hspace{-0.1cm}  \in   \hspace{-0.1cm} \Big[ \hspace{-1pt}   \f{g(h)y}{2}  \hspace{-1pt} , g(h)(y-\kappa) \hspace{-1pt} \Big]   \hspace{-1.5pt};  \hspace{-0.05cm}   \cal{O}_h   \hspace{-3pt}    \r)
  \\
  \nonumber
  &+      \pp\l(   X_h \in g(h) dy ;       \Delta_1^{g(h)}   \leq h ;       X_{\Delta_1^{g(h)}  -}   < g(h)       ;  S_{\Delta_1^{g(h)}}  \geq   g(h)(y-\kappa)    ; \cal{O}_h    \r)
  \\
  \label{3a3b}
  &=:          \sigma_h^{3a}(dy)  +       \sigma_h^{3b}(dy) .
  \end{align}

    \p{Upper Bound for \mbox{\hyperref[3a3b]{$\sigma_h^{3a}(dy)$}}} Disintegrating on the values of $\Delta_1^{g(h)} $, $ X_{\Delta_1^{g(h)}-} $ and  $S_{\Delta_1^{g(h)}}  $,   then applying the Markov property,
     $\l(\ref{l6cond}\r)$ and Lemma \mbox{\ref{bigjump}}, it follows that uniformly among $y>K$ as $h\to\infty$, with $\pp(X_t\in dx) \hspace{-2pt} = \hspace{-2pt} f_t(x)dx  $,
    \b{align*}
  \mbox{\hyperref[3a3b]{$\sigma_h^{3a}(dy)$}} &\overset{\phantom{\l(\ref{l6cond}\r)}}=     \int_{s=0}^h \int_{w=0}^1 \int_{v=\f{y}{2}}^{y-\kappa}  \pp\Big(   X_h \in g(h) dy ;   \Delta_1^{g(h)}   \in ds  ;
  \\
  &\ \hspace{3.4cm} \hspace{3pt}     X_{\Delta_1^{g(h)}  -}   \in  g(h)dw     ;   S_{\Delta_1^{g(h)}}   \in   g(h)dv      ;\cal{O}_h    \Big)
\\
  &\overset{\phantom{\l(\ref{l6cond}\r)}}=  \int_0^h \int_0^1 \int_{\f{y}{2}}^{y-\kappa} f_{h-s}  ( g(h) (y-w-v) ) g(h)dy  \pp\l(  S_{\Delta_1^{g(h)}}   \in   g(h)dv     \r)
  \\
  &\ \hspace{2.5cm} \times \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r)

   \\
     &\overset{\l(\ref{l6cond}\r)}\lesssim    \int_0^h \int_0^1 \int_{\f{y}{2}}^{y-\kappa}    \f{ (h-s)  L(g(h)(y-w-v)    )    }{g(h)^{\alpha} (y-w-v)^{1+\alpha} }  dy  \pp\l(  S_{\Delta_1^{g(h)}}   \in   g(h)dv     \r)
\\
  &\ \hspace{2.5cm}   \times
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r)
     \\
     &\hspace{1pt}  \overset{\ref{bigjump}}\lesssim    \int_0^h \int_0^1 \int_{\f{y}{2}}^{y-\kappa}    \f{ (h-s)  L(g(h)(y-w-v)    )    }{g(h)^{\alpha} (y-w-v)^{1+\alpha} }  dy   v^{-1-\alpha} \f{L(g(h)v)}{L(g(h))} dv
\\
  &\ \hspace{2.5cm}   \times
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r).

   \intertext{As $ y- \kappa = y- x_0/g(h)-2  \leq y- x_0/g(h)-w$, $y/2 \geq y/2-w$, and $h-s\leq h$, }

     \mbox{\hyperref[3a3b]{$\sigma_h^{3a}(dy)$}}   &\lesssim    \int_0^h \int_0^1 \int_{\f{y}{2} -w}^{y- \f{x_0}{g(h)}-w}    \f{h    L(g(h)(y-w-v)    )   }{   g(h)^{\alpha} (y-w-v)^{1+\alpha}   v^{1+\alpha} }  dy  \f{L(g(h)v)}{L(g(h))} dv
     \\
     &\ \hspace{3.25cm}
     \times
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r).

   \intertext{Changing variables from $v$ to $z=v+w$, then from $z$ to $\rho= z/y$,}

     \mbox{\hyperref[3a3b]{$\sigma_h^{3a}(dy)$}}   &\lesssim    \int_0^h \int_0^1 \int_{\f{y}{2}}^{y- \f{x_0}{g(h)}}    \f{  h     L(g(h)(y-z)    )   }{  g(h)^{\alpha} (y-z)^{1+\alpha} (z-w)^{1+\alpha}   }  dy   \f{L(g(h)(z-w))}{L(g(h))}  dz
\\
     &\ \hspace{3cm}     \times
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r)
 \\
   &=    \int_0^h \int_0^1 \int_{\f{1}{2}}^{1- \f{x_0}{g(h)y}}      \hspace{-0.75cm}
    \f{ h     L(g(h)y (1-\rho)    )    }{   g(h)^{\alpha} y^{1+\alpha} (1-\rho)^{1+\alpha}  (y\rho-w)^{1+\alpha}   }  dy  \f{L(g(h)(y\rho -w))}{L(g(h))}  y d\rho
    \\
          &\ \hspace{2.25cm}    \times
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r).

   \intertext{As  $L$ is \mbox{\hyperref[sv]{slowly varying}} at $\infty$ and   $y   \geq   y\rho   -   w     \geq      y/4$,   by the uniform convergence theorem \cite[Theorem 1.2.1]{bgt89}, $(y\rho-w)^{-1-\alpha}  L(g(h)(y\rho -w))    \lesssim   y^{-1-\alpha}  L(g(h)y) $  uniformly in    $y   >    K$ as $h   \to   \infty$,   and so we have  uniformly in    $y   >    K$ as $h   \to   \infty$, }

   \mbox{\hyperref[3a3b]{$\sigma_h^{3a}(dy)$}} &\lesssim     \int_0^h \int_0^1 \int_{\f{1}{2}}^{1- \f{x_0}{g(h)y}}    \f{ h   L(g(h)y (1-\rho)    )   }{  g(h)^{\alpha} y^{1+\alpha} (1-\rho)^{1+\alpha}   y^{\alpha}     }    dy    \f{L(g(h)y)}{L(g(h))} d\rho
\\
    &\ \hspace{3cm}
     \times
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r).

   \intertext{Changing variables from $\rho$ to $\tau=1-\rho$, then from $\tau$ to $\chi= g(h) y \tau$, }

   \mbox{\hyperref[3a3b]{$\sigma_h^{3a}(dy)$}} &\lesssim     \int_0^h \int_0^1 \int_{ \f{x_0}{g(h)y}}^{\f{1}{2}}     \f{  h     L(g(h)y \tau    )      }{ g(h)^{\alpha} y^{1+\alpha} \tau^{1+\alpha}    y^{\alpha}   }     dy   \f{L(g(h)y)}{L(g(h))} d\tau
  \\
      &\ \hspace{2.5cm}
     \times
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r)
\\
&=    \int_0^h \int_0^1 \int_{ x_0}^{\f{g(h)y}{2}}     \f{   h    L(\chi   )     }{  \chi^{1+\alpha}   y^{1+\alpha}    }  dy   \f{L(g(h)y)}{L(g(h))} d\chi
   \\
       &\ \hspace{2.6cm}
     \times
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r)
\\
&= y^{-1-\alpha}     \f{L(g(h)y)}{ L(g(h))  }  dy   \int_0^h \int_0^1 \int_{ x_0}^{\f{g(h)y}{2}}  \f{ h L(\chi   )   }{  \chi^{1+\alpha}}    d\chi
  \\
      &\ \hspace{4.75cm}
     \times
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r).
\end{align*}

   Recall that $\ov$ is \mbox{\hyperref[rv]{regularly varying}} at $\infty$, $\ov(x) = x^{-\alpha} L(x)$, and $u(x)dx=\Pi(dx)$. One can   verify that  $\ov(x) \asymp x u(x)$ for all large enough $x$, see  \cite[Prop 2.1.1]{bgt89}.     Then by the condition $\l(\ref{l6cond}\r)$, having chosen $x_0$   large enough that    $\ov(x) \asymp x u(x)$,
   \[  \int_{ x_0}^{\f{g(h)y}{2}} \f{h    L(\chi   ) }{  \chi^{1+\alpha}}  d\chi   \asymp     \int_{ x_0}^{\f{g(h)y}{2}} h    u(\chi   )    d\chi       \overset{\l(\ref{l6cond}\r)}\lesssim  \pp\l( X_{   h    }  \in \l[x_0,  \f{g(h)y}{2}\r]  \r)
   \leq 1,
   \]
   and so we   conclude, as required, that uniformly in $y>K$, as $h\to\infty$,
    \b{align} \label{3aupperbound}   \nonumber
    \mbox{\hyperref[3a3b]{$\sigma_h^{3a} (dy)$}} &\lesssim    y^{-1-\alpha}     \f{L(g(h)y)}{ L(g(h))  }  dy   \int_0^h \int_0^1
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r)
\\
  &\leq   y^{-1-\alpha}     \f{L(g(h)y)}{ L(g(h))  }  dy  \ \pp\l(    \Delta_1^{g(h)}  \leq h  ;       X_{\Delta_1^{g(h)}  -}  < g(h)        ;\cal{O}_h    \r)
\\      \label{penultimatebound}
 &\leq   y^{-1-\alpha}     \f{L(g(h)y)}{ L(g(h))  }   \pp(\cal{O}_h) dy .
 \end{align}

 Now we consider \mbox{\hyperref[3a3b]{$   \sigma_h^{3b}(dy) $}}. To prove the desired bound for $   \sigma_h^{3b}(dy) $, first we bound the density  $  \sigma_h^{3b}(dy)$, and then we bound the whole integral $\int_K^{w(h)}   q_h(yg(h))   \sigma_h^{3b}(dy) $.
      \p{Upper Bound for \mbox{\hyperref[3a3b]{$\sigma_h^{3b}(dy)$}}}  By $\l(\ref{contain}\r)$,   disintegrating on the value of $S_{\Delta_1^{g(h)}}$,
      \b{align}
      \nonumber
      \mbox{\hyperref[3a3b]{$\sigma_h^{3b}(dy)$}} &\overset{\phantom{\l(\ref{contain}\r)}}{:=}  \pp\l(   X_h \in g(h) dy ;      X_{T_{g(h)}} >  \f{g(h)y}{2}      ;  S_{\Delta_1^{g(h)}}  \geq   g(h)(y-\kappa)    ; \cal{O}_h    \r)
      \\
      \nonumber
      &\overset{\l(\ref{contain}\r)}=  \hspace{-2pt}     \pp  \hspace{-2pt}     \l( \hspace{-4pt}  X_h \hspace{-2pt} \in \hspace{-2pt} g(h) dy ; \Delta_1^{g(h)}
     \hspace{-4pt}  =  \hspace{-2pt}  T_{g(h)}  \hspace{-3pt}  \leq  \hspace{-3pt}  h      ;  \hspace{-1pt}   X_{\Delta_1^{g(h)} \hspace{-3pt}  -} \hspace{-6pt} < \hspace{-2pt}    g(h)        ;  \hspace{-1pt} S_{\Delta_1^{g(h)}} \hspace{-2pt} \geq  \hspace{-2pt} g(h)(y \hspace{-1pt}  - \hspace{-1pt}  \kappa)    ; \hspace{-1pt} \cal{O}_h  \hspace{-2.5pt}  \r)
   \\
   \nonumber
     &\overset{\phantom{\l(\ref{contain}\r)}}= \hspace{-6pt}     \int_{      y-  \kappa  }^{y}  \hspace{-6pt}      \pp  \hspace{-1pt}   \l(   \hspace{-2pt}   X_h \in g(h) dy ; \Delta_1^{g(h)} \leq h      ;   X_{\Delta_1^{g(h)} -} < g(h)       ;  S_{\Delta_1^{g(h)}}\in  g(h)dv   ; \cal{O}_h      \hspace{-2pt}   \r) \hspace{-2pt}  .

  \intertext{Conditionally given $\Delta_1^{g(h)}  \leq h$ and   $S_{\Delta_1^{g(h)}} \in g(h)dv$, by the Markov property,   $X_h \overset{d}=X_{\Delta_1^{g(h)}-} +  \hat{X}_{h-\Delta_1^{g(h)}} +   g(h)v$, with $\hat{X}$   an independent copy of $X$,   so}

 \nonumber
      \mbox{\hyperref[3a3b]{$\sigma_h^{3b}(dy)$}}   &=        \int_{      y-  \kappa  }^{y}        \pp\Big(  X_{\Delta_1^{g(h)}-} +  \hat{X}_{h-\Delta_1^{g(h)}}  +  g(h)v \in g(h) dy ; \Delta_1^{g(h)} \leq h      ;
    \\
    \nonumber
     &\ \hspace{1.75cm}
  X_{\Delta_1^{g(h)} -}  < g(h)       ;  S_{\Delta_1^{g(h)}}\in  g(h)dv   ; \cal{O}_h    \Big).

  \intertext{Note that $\cal{O}_h$ is  attained by time $\Delta_1^{g(h)}$,  and $x^{-\alpha} L(x)=\ov(x)$ is non-increasing. By  the Markov property, then by    Lemma \mbox{\ref{bigjump}}, as $h\to\infty$, uniformly in $y>K$,}

\nonumber
     \mbox{\hyperref[3a3b]{$\sigma_h^{3b}(dy)$}}   &\overset{\phantom{\ref{bigjump}}}\leq    \int_{      y-  \kappa }^{y}        \pp\Big(    X_{\Delta_1^{g(h)}-} +  \hat{X}_{h-\Delta_1^{g(h)}} + g(h) v \in g(h) dy ; \Delta_1^{g(h)} \leq h      ;
    \nonumber
      \\
     &\ \hspace{1.75cm}  X_{\Delta_1^{g(h)} -} < g(h)          ; \cal{O}_h    \Big)   \pp\l( S_{\Delta_1^{g(h)}}\in  g(h)dv\r)
\nonumber
  \\
  \nonumber
     &\overset{\ref{bigjump}}\lesssim        \int_{      y-   \kappa }^{y}        \pp\Big(         X_{\Delta_1^{g(h)}-} +  \hat{X}_{h-\Delta_1^{g(h)}}         + g(h) v \in g(h) dy ; \Delta_1^{g(h)} \leq h   ;
     \\
     \nonumber  &\ \hspace{1.75cm}    X_{\Delta_1^{g(h)} -}   < g(h)          ; \cal{O}_h    \Big)
  v^{-1-\alpha}    \f{L(g(h)v)}{L(g(h))} dv
\nonumber
\\
\nonumber
  &\overset{\phantom{\ref{bigjump}}}\leq     \f{L\l(g(h)  \l(   y-   \kappa  \r)  \r)}{   \l(  y-   \kappa  \r)^{1+\alpha}  L(g(h))}   \int_{      y- \kappa }^{y}        \pp\Big(    X_{\Delta_1^{g(h)}-} +  \hat{X}_{h-\Delta_1^{g(h)}}  + g(h) v \in g(h) dy ;
    \\
    \nonumber
  &\  \hspace{4.75cm} \hspace{2pt}
   \Delta_1^{g(h)} \leq h      ;   X_{\Delta_1^{g(h)} -}  < g(h)          ; \cal{O}_h    \Big)   dv.

  \intertext{Now, $L$ is \mbox{\hyperref[sv]{slowly varying}} at $\infty$ and $\lim_{h\to\infty}\kappa    =    2$, so by the uniform convergence theorem \cite[Theorem 1.2.1]{bgt89}, uniformly in $y     >     K$~as~$h   \to   \infty$, }

\nonumber
     \mbox{\hyperref[3a3b]{$\sigma_h^{3b}(dy)$}}  &\lesssim     y^{-1-\alpha}     \f{L(g(h)    y)}{L(g(h))}    \int_{      y- \kappa }^{y}        \pp\Big(   X_{\Delta_1^{g(h)}-} +  \hat{X}_{h-\Delta_1^{g(h)}}  + g(h) v \in g(h) dy ;
       \\
       \nonumber
  &\  \hspace{4.25cm}    \Delta_1^{g(h)} \leq h      ;   X_{\Delta_1^{g(h)} -}  < g(h)          ; \cal{O}_h
    \Big)   dv
    \\
    \label{EE}
    &=:      y^{-1-\alpha}     \f{L(g(h)    y)}{L(g(h))}    \int_{      y- \kappa }^{y}        \pp (
    E_{h,v,dy})dv.
\end{align}

   \p{Bounding the Integral}   Now we will bound the whole integral for \mbox{\hyperref[3a3b]{$\sigma_h^{3b}(dy)$}},
\b{equation}
   \label{boundingtheintegral}
   \int_K^{w(h)}    q_h(yg(h))            \sigma_h^{3b}(dy) .
\end{equation}
By $\l(\ref{EE}\r)$, writing our integral   as a sum of integrals over unit intervals,
   \b{align*}
  \l(\ref{boundingtheintegral}\r)     &\overset{\l(\ref{EE}\r)}\lesssim           \int_K^{w(h)}    q_h(yg(h))        y^{-1-\alpha}    \f{L(g(h)    y)}{L(g(h))}      \int_{      y- \kappa  }^{y}        \pp\l(
     E_{h,v,dy}
   \r)   dv
\\
  &\overset{\phantom{\l(\ref{EE}\r)}}\leq     \sum_{k=\lfloor K\rfloor}^{\lfloor w(h) \rfloor }    \int_{y=k}^{k+1}     q_h(yg(h))        y^{-1-\alpha}    \f{L(g(h)    y)}{L(g(h))}      \int_{      y- \kappa  }^{y}        \pp\l(
   E_{h,v,dy}
     \r)   dv.
\intertext{By Theorem \mbox{\ref{recurrentthm}},  $q_h(\cdot)$ is non-decreasing, then since $L$ is \mbox{\hyperref[sv]{slowly varying}}  at $\infty$, applying the uniform convergence theorem \cite[Theorem 1.2.1]{bgt89},}

  \l(\ref{boundingtheintegral}\r)     &\lesssim       \sum_{k=\lfloor K\rfloor}^{\lfloor w(h) \rfloor }    \int_{y=k}^{k+1}     q_h((k+1)g(h))        k^{-1-\alpha}     \f{L(g(h)    k)}{L(g(h))}     \int_{      y- \kappa  }^{y}        \pp\l(
    E_{h,v,dy}
     \r)   dv.

  \intertext{Exchanging the order of integration,}

    \l(\ref{boundingtheintegral}\r)     &\lesssim           \sum_{k=\lfloor K\rfloor}^{\lfloor w(h) \rfloor }       q_h((k+1)g(h))        k^{-1-\alpha}    \f{L(g(h)    k)}{L(g(h))}         \int_{ v=k -  \kappa }^{k+1}             \int_{y=v\vee k}^{( v+   \kappa ) \wedge (k+1)  }   \hspace{-10pt}  \pp\l(
      E_{h,v,dy}
       \r)   dv
\\
     &\leq        \sum_{k=\lfloor K\rfloor}^{\lfloor w(h) \rfloor }       q_h((k+1)g(h))        k^{-1-\alpha}     \f{L(g(h)    k)}{L(g(h))}        \int_{ v=k -  \kappa }^{k+1}             \int_{y=v}^{ v+   \kappa  }     \pp\l(
      E_{h,v,dy}
       \r)   dv.

 \intertext{Now, define the event    $A_h:= \{ \Delta_1^{g(h)} \leq h      ;   X_{\Delta_1^{g(h)} -}  < g(h)          ; \cal{O}_h     \}$, so that
 \[
      \int_{y=v}^{ v+   \kappa  }     \pp\l(
      E_{h,v,dy}
       \r)
   =     \int_{y=v}^{ v+   \kappa  }     \pp\l(   X_{\Delta_1^{g(h)}-} +  \hat{X}_{h-\Delta_1^{g(h)}}    + g(h) v \in g(h) dy ;
      A_h
       \r)
  \]
  \[
   \hspace{1.95cm} =      \pp\l(   X_{\Delta_1^{g(h)}-} +  \hat{X}_{h-\Delta_1^{g(h)}}    \in  [0, g(h) \kappa] ;
      A_h
       \r) =: \pp(U_h) .
  \]

Observing that     $

       \pp(U_h)  $ is independent of~$v$~and~$k$, and $\lim_{h\to\infty}\kappa= 2$, as $h\to\infty$,}

    \l(\ref{boundingtheintegral}\r)     &\lesssim

     \sum_{k=\lfloor K\rfloor}^{\lfloor w(h) \rfloor }       q_h((k+1)g(h))        k^{-1-\alpha}        \f{L(g(h)    k)}{L(g(h))}      \pp(U_h)      \int_{ v=k - \kappa }^{k+1}                        dv
       \\
  &=

      \sum_{k=\lfloor K\rfloor}^{\lfloor w(h) \rfloor }       q_h((k+1)g(h))        k^{-1-\alpha}    \f{L(g(h)    k)}{L(g(h))}    \pp(U_h)       \l(1 + \kappa \r).

\\

      &\lesssim

    \sum_{k=\lfloor K\rfloor}^{\lfloor w(h) \rfloor }       q_h((k+1)g(h))        k^{-1-\alpha}   \f{L(g(h)    k)}{L(g(h))}     \pp(U_h)    .

\\
&=

   \sum_{k=\lfloor K\rfloor}^{\lfloor w(h) \rfloor }       \int_{y=k+1}^{k+2}       q_h((k+1)g(h))        k^{-1-\alpha}    \f{L(g(h)    k)}{L(g(h))}           \pp(U_h) dy.

  \intertext{Recall $q_h(\cdot)$ is non-decreasing by Theorem \mbox{\ref{recurrentthm}}. As $L$ is \mbox{\hyperref[sv]{slowly varying}} at $\infty$ and  $y \asymp k$ uniformly in $k>\lfloor K\rfloor $ and $y\in[k+1,k+2]$, it follows by the uniform convergence theorem \cite[Theorem 1.2.1]{bgt89} that  $L(g(h) k) \lesssim L(g(h)y)$ uniformly in  $k>\lfloor K\rfloor $ and $y\in[k+1,k+2]$ as $h\to\infty$, and therefore}

    \l(\ref{boundingtheintegral}\r)     &\overset{\ref{recurrentthm}}\lesssim

 \sum_{k=\lfloor K\rfloor}^{\lfloor w(h) \rfloor }     \int_{y=k+1}^{k+2}        q_h(yg(h))        k^{-1-\alpha}    \f{L(g(h)    k)}{L(g(h))}    \pp(U_h)     dy
\\
  &\overset{\phantom{\ref{recurrentthm}}}\lesssim

        \sum_{k=\lfloor K\rfloor}^{\lfloor w(h) \rfloor }     \int_{y=k+1}^{k+2}          q_h(yg(h))        y^{-1-\alpha}    \f{L(g(h)    y)}{L(g(h))}          \pp(U_h)   dy
      \\
       &\overset{\phantom{\ref{recurrentthm}}}\leq

            \int_{y=K}^{\infty}          q_h(yg(h))        y^{-1-\alpha}    \f{L(g(h)    y)}{L(g(h))}      \pp(U_h)      dy.

  \end{align*}
  Now,  $U_h\subseteq  A_h =  \{ \Delta_1^{g(h)} \leq h      ;   X_{\Delta_1^{g(h)} -}  < g(h)          ; \cal{O}_h     \}     \subseteq \cal{O}_h$, so   as $h\to\infty$,

  \b{align}
    \label{3bupperbound}
    \l(\ref{boundingtheintegral}\r)     &\overset{\phantom{\ref{recurrentthm}}}\lesssim    \hspace{-3pt}    \pp  \hspace{-1pt}  \l(  \hspace{-1pt}   \Delta_1^{g(h)} \hspace{-1pt}  \leq  \hspace{-1pt}  h      ;    \hspace{-1pt}  X_{\Delta_1^{g(h)} -}   \hspace{-3pt}  <  \hspace{-1pt}  g(h)          ;  \hspace{-1pt}  \cal{O}_h      \hspace{-1pt}
       \r)  \hspace{-3pt}   \int_{y=K}^\infty   \hspace{-2pt}        q_h(yg(h))        y^{-1-\alpha}    \f{L(g(h)    y)}{L(g(h))}   dy
\\
\label{finalbound}
  &\overset{\phantom{\ref{recurrentthm}}}\leq      \int_{y=K}^\infty         q_h(yg(h))        y^{-1-\alpha}    \f{L(g(h)    y)}{L(g(h))}    \pp\l(  \cal{O}_h    \r) dy.
  \end{align}

      \end{comment}

   \b{comment}
   \p{Summary of Upper Bounds}    We have shown that as $h\to\infty$,
   \[
   \int_K^\infty    q_h(y g(h))     \sigma_h(dy) \lesssim   o(1) \times \pp(\cal{O}_h)   \int_{y=\lfloor{K}\rfloor}^\infty         q_h(yg(h))        y^{-1-\alpha}     dy
   \]
   \[
   +     2   \pp\l(   \Delta_1^{g(h)} \leq h      ;   X_{\Delta_1^{g(h)} -}  < g(h)          ; \cal{O}_h
       \r)   \int_{y=\lfloor{K}\rfloor}^\infty         q_h(yg(h))        y^{-1-\alpha}  \f{L(g(h)    y)}{L(g(h))}    dy
    \]
   \[
   \lesssim  \pp\l( \cal{O}_h   \r) \int_{y=\lfloor{K}\rfloor}^\infty         q_h(yg(h))        y^{-1-\alpha}    \f{L(g(h)    y)}{L(g(h))}    dy.
   \]
  \end{comment}

   \p{Upper Bound for \mbox{\hyperref[starting]{$\sigma_h^3(dy)$}}}     Disintegrating on the values of $\Delta_1^{g(h)} $, $ X_{\Delta_1^{g(h)}-} $ and  $S_{\Delta_1^{g(h)}}  $,   then applying the Markov property

  and Lemma \mbox{\ref{bigjump}}, it follows that uniformly among $y>K$ as $h\to\infty$, with $\pp(X_t\in dx) \hspace{-2pt} = \hspace{-2pt} f_t(x)dx  $,
    \b{align}
        \nonumber
  \mbox{\hyperref[starting]{$\sigma_h^{3}(dy)$}} &\overset{\phantom{\ref{bigjump}}}=     \int_{s=0}^h \int_{w=0}^1 \int_{v=\f{y}{2}}^{y-w}  \pp\Big(   X_h \in g(h) dy ;   \Delta_1^{g(h)}   \in ds  ;
  \\
        \nonumber
  &\ \hspace{3.4cm} \hspace{3pt}     X_{\Delta_1^{g(h)}  -}   \in  g(h)dw     ;   S_{\Delta_1^{g(h)}}   \in   g(h)dv      ;\cal{O}_h    \Big)
\\
        \nonumber
  &\overset{\phantom{\ref{bigjump}}}=  \int_0^h \int_0^1 \int_{\f{y}{2}}^{y-w} f_{h-s}  ( g(h) (y-w-v) ) g(h)dy  \pp\l(  S_{\Delta_1^{g(h)}}   \in   g(h)dv     \r)
  \\
        \nonumber
  &\ \hspace{2.5cm} \times \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r)

  \end{align}
\begin{align}
        \nonumber
     &   \overset{\ref{bigjump}}\lesssim    \int_0^h \int_0^1 \int_{\f{y}{2}}^{y-w}      f_{h-s}  ( g(h) (y-w-v) ) g(h)dy               \  v^{-1-\alpha}  \f{L(g(h)v)}{    L(g(h))} dv
\\
        \nonumber
  &\ \hspace{2.5cm}   \times
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r)

      \\
        \nonumber
      &\overset{\phantom{\ref{bigjump}}}\leq    \int_0^h \int_0^1 \int_{\f{y}{2} -w}^{y- w}    f_{h-s}   ( g(h) (y-w-v) ) g(h) dy \   v^{-1-\alpha}   \f{L(g(h)v)}{     L(g(h))} dv
     \\
        \nonumber
     &\ \hspace{2.5cm}
     \times
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r).

\intertext{Now, as $ y/3 \leq y/2-1\leq  y/2 -w \leq v \leq y$, applying the uniform convergence theorem \cite[Theorem 1.2.1]{bgt89} to $L(g(h)v)$,  uniformly in $y>K$ as $h\to\infty$,}

\mbox{\hyperref[starting]{$\sigma_h^{3}(dy)$}} &\lesssim y^{-1-\alpha}   \f{L(g(h)y)}{     L(g(h))} dy    \int_0^h \int_0^1 \int_{\f{y}{2} -w}^{y- w}    f_{h-s}   ( g(h) (y-w-v) ) g(h)   dv
     \\
        \nonumber
     &\ \hspace{4.4cm}
     \times
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r).

   \intertext{Changing variables to $u  =   g(h)(y-w-v)$,    uniformly in~$y  >  K$,~as~$h  \to  \infty$, }

        \nonumber
     \mbox{\hyperref[starting]{$\sigma_h^{3}(dy)$}}   &\lesssim  y^{-1-\alpha}    \f{L(g(h)y)}{      L(g(h))} dy    \int_0^h \int_0^1 \int_{0}^{\f{g(h)y}{2} }   f_{h-s}  ( u ) du
\\
        \nonumber
     &\ \hspace{4.4cm}     \times
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r)
   \\
        \nonumber
   &=  y^{-1-\alpha}   \f{L(g(h)y)}{      L(g(h))} dy    \int_0^h \int_0^1    \pp\l(  X_{h-s} \leq \f{g(h)y}{2} \r)
\\
        \nonumber
     &\ \hspace{4.4cm}     \times
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r)
        \\
        \nonumber
   &\leq  y^{-1-\alpha}   \f{L(g(h)y)}{      L(g(h))} dy    \int_0^h \int_0^1
      \pp\l(    \Delta_1^{g(h)}   \in ds  ;       X_{\Delta_1^{g(h)}  -}   \in  g(h)dw        ;\cal{O}_h    \r)
     \\
       \label{penultimatebound}
         &=  y^{-1-\alpha}   \f{L(g(h)y)}{      L(g(h))} dy
      \pp\l(    \Delta_1^{g(h)}   \leq h  ;       X_{\Delta_1^{g(h)}  -}   < g(h)       ;\cal{O}_h    \r)
          \\
            \label{sigma3bound}
         &\leq  y^{-1-\alpha}    \f{L(g(h)y)}{    L(g(h))}     \pp\l(    \cal{O}_h    \r) dy
    .

\end{align}

    \p{Conclusion of Upper Bound} By

   $\l(\ref{starting}\r)$,   $\l(\ref{sigma1bound}\r)$, $\l(\ref{sigma2bound}\r)$, and $\l(\ref{sigma3bound}\r)$,          we conclude, as required for the upper bound in $\l(\ref{l6statement}\r)$, that  uniformly in $y>K$,    as $h\to\infty$,
   \b{equation} \label{sigmaupper}
   \pp(X_h \in g(h) dy    ;   \cal{O}_h ) \lesssim     y^{-1-\alpha}  \f{  L(g(h)y)}{ L(g(h))}   \pp(\cal{O}_h)  dy.
     \end{equation}

 Now we will prove the lower bound  on $  \pp\l(       X_h \in g(h)dy   ; \cal{O}_h  \r) $.

\p{Proof of Lower Bound}   Now, fixing  $y_0>0$,     for  all  $y>K$, as $h\to\infty$,
\begin{align}
\nonumber
     \pp\l(       X_h \in g(h)dy   ; \cal{O}_h  \r)    &\geq   \pp\Big(       X_h \in g(h)dy   ;      \D^{g(h)}   \leq h-h_0   ;   X_{\Delta_1^{g(h)}-}   < g(h)    ;    \hspace{3cm}
\\
     \label{lb}
   &\hspace{3.25cm}         \f{g(h)y}{2}  \leq S_{\Delta_1^{g(h)}}\leq    g(h)y-y_0      ;    \cal{O}_h   \hspace{-1pt}  \Big)              .
\end{align}

Disintegrating on the values of $  \Delta_1^{g(h)} $, $ X_{\Delta_1^{g(h)}-}$, and $S_{\Delta_1^{g(h)}}  $,       applying the Markov property, noting that by $\l(\ref{s-}\r)$,  $\{ \Delta_1^{g(h)} \hspace{-2pt}  \in  \hspace{-2pt} ds ;  \cal{O}_h \} \hspace{-2pt}=\hspace{-2pt} \{ \Delta_1^{g(h)} \hspace{-2pt}  \in  \hspace{-2pt} ds ;  \cal{O}_{\Delta_1^{g(h)}} \}$  for each $s\leq h$,

with~$\pp(X_t \in dx) \hspace{-2pt}=\hspace{-2pt} f_t(x)dx$,

\b{align}
   \nonumber
    \l(\ref{lb}\r)

    &=       \int_{s=0}^{h-1}   \int_{w=0}^1   \int_{v=\f{y}{2}}^{y-\f{y_0}{g(h)}-w }      \pp\Big(    X_{h} \in g(h)dy ;   \Delta_1^{g(h)} \in ds  ;   X_{\Delta_1^{g(h)}-}   \in g(h) dw   ;
    \\
    \nonumber
        &\ \hspace{4.5cm}    S_{\Delta_1^{g(h)}}  \in g(h) dv    ; \cal{O}_{\Delta_1^{g(h)}}      \Big)
        \\
    \nonumber
     & =           \int_{0}^{h-1}   \int_{0}^1   \int_{v=\f{y}{2}}^{y-\f{y_0}{g(h)}-w}   f_{h-s}(g(h)(y-w-v))  g(h) dy  \pp\l( S_{\Delta_1^{g(h)}}  \in g(h) dv        \r)
   \\
    \nonumber
    &\ \hspace{3.25cm} \times \pp\l(   \Delta_1^{g(h)} \in ds  ;   X_{\Delta_1^{g(h)}-}   \in g(h) dw   ; \cal{O}_{h}  \r) .

  \intertext{Applying Lemma \mbox{\ref{bigjump}},  noting    $h-s\geq1$,   $y/2 <  2y/3 -w$, $v\asymp y$, and    $   L(g(h)v)\hspace{-1pt}  \asymp \hspace{-1pt}    L(g(h)y)$ uniformly in $y>K$ as $h\to\infty$ by the uniform convergence theorem \cite[Thm 1.2.1]{bgt89}, it follows that uniformly in~$y  >  K$~as~$h  \to  \infty$,  }

   \nonumber
     \l(\ref{lb}\r)   &\overset{\ref{bigjump}}\gtrsim          \int_{0}^{h-1}   \int_{0}^1   \int_{v=\f{y}{2}}^{y-\f{y_0}{g(h)} - w}    f_{h-s} (g(h)(y-w-v) ) g(h)   dy        v^{-1-\alpha}  \f{L(g(h)v)}{L(g(h))}        dv
   \\
    \nonumber
    &\ \hspace{3.25cm} \times \pp\l(   \Delta_1^{g(h)} \in ds  ;   X_{\Delta_1^{g(h)}-}   \in g(h) dw   ; \cal{O}_h  \r)

  \\
    \nonumber
     &\gtrsim          \int_{0}^{h-1}   \int_{0}^1   \int_{v=\f{2y}{3}-w}^{y-\f{y_0}{g(h)} - w}    f_{h-s} (g(h)(y-w-v) )     g(h)   dy        y^{-1-\alpha}     \f{L(g(h)y)}{L(g(h))}          dv
   \\
    \nonumber
  &\ \hspace{3.25cm} \times \pp\l(   \Delta_1^{g(h)} \in ds  ;   X_{\Delta_1^{g(h)}-}   \in g(h) dw   ; \cal{O}_h  \r) .

       \intertext{Changing   variables   to $u =    g(h) (y -w-v)  $, noting   $y/3 > 1$ for all  $y>K$,
    }

   \nonumber
             \l(\ref{lb}\r)        &\gtrsim

       y^{-1-\alpha}      \f{L(g(h)y)}{L(g(h))}         dy   \int_{0}^{h-1}   \int_{0}^1          \int_{u=y_0}^{\f{g(h)y}{3 } }      f_{h-s}(u) du
   \\
   \nonumber
  &\ \hspace{4.35cm} \times \pp\l(   \Delta_1^{g(h)} \in ds  ;   X_{\Delta_1^{g(h)}-}   \in g(h) dw   ; \cal{O}_h  \r)

   \\
  \nonumber
  &=      y^{-1-\alpha}      \f{L(g(h)y)}{L(g(h))}         dy   \int_{0}^{h-1}   \int_{0}^1    \l[     \pp\l(  X_{h-s}\leq  \f{g(h)y}{3}    \r) -     \pp\l(  X_{h-s}\leq y_0    \r) \r]
   \\
   \nonumber
  &\ \hspace{4.35cm} \times \pp\l(   \Delta_1^{g(h)} \in ds  ;   X_{\Delta_1^{g(h)}-}   \in g(h) dw   ; \cal{O}_h  \r)
   \\
  \nonumber
  &\geq      y^{-1-\alpha}      \f{L(g(h)y)}{L(g(h))}         dy   \int_{0}^{h-1}   \int_{0}^1   \l[     \pp\l(  X_{h}\leq g(h)    \r) -     \pp\l(  X_{1}\leq y_0    \r)   \r]
   \\
   \nonumber
  &\ \hspace{4.35cm} \times \pp\l(   \Delta_1^{g(h)} \in ds  ;   X_{\Delta_1^{g(h)}-}   \in g(h) dw   ; \cal{O}_h  \r).
  \end{align}

Now,  with $X^{(0,x)}$ again denoting the process with no jumps bigger than $x$,
  \[
  \pp\l(   X_h \leq    g(h) \r)  =   \pp(   X_h^{(0,g(h))} \leq    g(h) ) \pp( \Delta_1^{g(h)} > h) =   \pp(   X_h^{(0,g(h))} \leq    g(h) ) e^{- h \ov(g(h))}  ,
  \]
  and   since $\lim_{h\to\infty} h \ov(g(h))=0$ by $\l(\ref{condt}\r)$, by Markov's inequality,  as $h\to\infty$,
  \[
 \pp\l(   X_h \leq    g(h) \r)    \overset{\l(\ref{condt}\r)}\sim  \      \pp(   X_h^{(0,g(h))} \leq    g(h) ) \   \geq 1 - \f{ \mathbb{E}[   X_h^{(0,g(h))}  ]   }{   g(h)    }    \geq      1 - \f{ h \int_0^{g(h)} \ov(x)dx   }{   g(h)    }.
  \]
  Now, by $\l(\ref{condt}\r)$ and Karamata's theorem  \cite[Prop 1.5.8]{bgt89}, as $h\to\infty$,
  \[
   \pp\l(   X_h \leq    g(h) \r)     \gtrsim        1    -  \f{    h g(h) \ov(g(h))    }{   g(h)   }   = \  1 - h \ov(g(h))  \  \overset{\l(\ref{condt}\r)}\sim  1.   \hspace{66pt}
  \]
  Then since $\pp(X_1\leq y_0) = \textup{constant} <1$,  uniformly in $y>K$ as $h\to\infty$,
  \begin{align}
   \nonumber
      \l(\ref{lb}\r)   &\gtrsim         \f{L(g(h)y)}{L(g(h))}         y^{-1-\alpha}      dy   \int_{s=0}^{h-1}   \int_{w=0}^1       \pp\l(   \Delta_1^{g(h)} \in ds  ;   X_{\Delta_1^{g(h)}-}   \in g(h) dw   ; \cal{O}_h  \r)

       \\
       \label{lowerbound}
       &=     \f{L(g(h)y)}{L(g(h))}        y^{-1-\alpha}      dy     \pp\l(    \Delta_1^{g(h)}      \leq h-1    ;   X_{\Delta_1^{g(h)}-}     <g(h)     ; \cal{O}_h    \r)   .
        \end{align}

    \b{comment}
       \p{Remark}  The powers highlighted in \textcolor{red}{red} are subject to some dispute/confusion. If we work at the level of densities, ignoring any potential $1-\rho$ term introduced by changing from $d(y-z)$ to $d((1-\rho))$, then we get the powers highlighted in red, and the argument works as above.

         If we do include the disputed $\textcolor{pink}{1-\rho}$ term in the above calculations, then since $\rho<1 - \f{1}{g(h)y} $ and $\lim_{h\to\infty} \f{1}{g(h)}= 0$, we can bound $1-\rho \geq   g(h) y \geq  20 g(h) \geq 20$, for all sufficiently large $h$, and the proof as above works.

         However, if we choose to include the disputed $\textcolor{pink}{1-\rho}$ term here, and also for the upper bound, then we end up with an upper bound which is strictly smaller than our lower bound, which is a contradiction.

         This fact alone confirms that we can ignore the $1-\rho$   term introduced by changing from $d(y-z)$ to $d((1-\rho)y)=(1-\rho)dy$, and so we should write the proof using densities rather than densities multiplied by $dy$.
        \end{comment}

\p{Proof by Contradiction Step}
Now we assume for a contradiction that
  \b{equation}
  \label{contradiction0}
  \liminf_{h\to\infty}     \f{  \pp\l(    \Delta_1^{g(h)} \leq h-1  ;   X_{\Delta_1^{g(h)}-}   <g(h)   ; \cal{O}_h    \r)    }{   \pp(\cal{O}_h)     }      = 0 .
   \end{equation}
   As    $\Delta_1^{g(h)} \hspace{-1.5pt} $ is exponentially distributed, rate $\ov(g(h)) \hspace{-0.5pt} $, by Lemma \mbox{\ref{lemma1}},   as~$h      \to      \infty \hspace{-0.5pt}$,
   \begin{align*}
  &  \pp\l(    \Delta_1^{g(h)} \in[ h-1 ,h] ;   X_{\Delta_1^{g(h)}-}   <g(h)   ; \cal{O}_h    \r)  \leq  \pp\l(    \Delta_1^{g(h)} \in[ h-1 ,h]       \r)
\\

   & \hspace{-4pt}\leq    \pp\l(    \Delta_1^{g(h)} \leq 1 \r)     \hspace{-1pt}   =    \hspace{-1pt}    1     \hspace{-1pt}    - e^{- \ov(g(h))}   \hspace{-1pt}    \leq     \hspace{-1pt}    \ov(g(h)) \overset{\ref{lemma1}}\sim   \f{\pp(\cal{O}_h )}{   \Phi(h)} \overset{\l(\ref{IfPhi}\r)}= o(1) \times \pp\l( \cal{O}_h\r)   ,
   \end{align*}
   since  $\lim_{h\to\infty} \Phi(h)=\infty$ by $\l(\ref{IfPhi}\r)$, so it follows that $\l(\ref{contradiction0}\r)$ holds if and only if
    \b{equation}
  \label{contradiction}
  \liminf_{h\to\infty}     \f{  \pp\l(    \Delta_1^{g(h)} \leq h  ;   X_{\Delta_1^{g(h)}-}   <g(h)   ; \cal{O}_h    \r)    }{   \pp(\cal{O}_h)     }      = 0 .
   \end{equation}

  Then by

  $\l(\ref{starting}\r)$,    $\l(\ref{sigma1bound}\r)$, $\l(\ref{sigma2bound}\r)$,  and $\l(\ref{penultimatebound}\r)$,   along some subsequence of $h$, as $h\to\infty$,

  \b{align}
\nonumber
  \pp  \l(   X_h    \geq      Kg(h) ;    \cal{O}_h    \r) &    =     \int_K^\infty        \pp(  X_h \in g(h) dy ; \cal{O}_h
     )      =       o(1)     \times       \pp(\cal{O}_h)        \int_{ K}^{ \infty}     y^{-1-\alpha  }       \f{L(g(h)y)}{L(g(h))}  dy      .

  \intertext{Changing variables from $y$ to $u=g(h)y$,  }

\nonumber
 \pp  \l(   X_h  \hspace{-2pt}  \geq    \hspace{-2pt}  Kg(h) ;    \cal{O}_h    \r) &  =   o(1) \times   \f{   g(h)^\alpha  \pp(\cal{O}_h)  }{     L(g(h))   }   \int_{Kg(h)}^{\infty}       u^{-1-\alpha  }       L(u)  du.

  \intertext{As $L$ is \mbox{\hyperref[sv]{slowly varying}} at $\infty$, applying the result   \cite[Prop 1.5.10]{bgt89}, as $h\to\infty$, }

  \pp\l(   X_h \geq K g(h) ;   \cal{O}_h    \r) &\lesssim     o(1) \times   \f{   g(h)^\alpha  \pp(\cal{O}_h)  }{       L(g(h))   }   (Kg(h))^{-\alpha}           L(Kg(h))

      \label{finall}

     = o(1) \times  \pp(\cal{O}_h)  .

 \intertext{But considering the subevent $ \{ \Delta_1^{g(h)}=\Delta_1^{Kg(h)}\leq h ; \cal{O}_h\}  \subseteq \{  X_h \geq K g(h) ;   \cal{O}_h     \}$,   disintegrating on the value of $\Delta_1^{g(h)} $, and applying  the Markov property,
}
   \nonumber
   \pp\l(   X_h \geq K g(h) ;   \cal{O}_h    \r) &\geq \pp\l(\Delta_1^{g(h)}=\Delta_1^{Kg(h)}\leq h ; \cal{O}_h\r)

 =     \int_0^h         \pp\l(\Delta_1^{g(h)}=\Delta_1^{Kg(h)}\in ds  ; \cal{O}_h\r)

  \\

   \nonumber
   &=      \int_0^h         \pp\l(\Delta_1^{g(h)} \in ds  ;    S_{\Delta_1^{g(h)}} \geq K g(h)      ; \cal{O}_{s,X^{(0,g(h))}}  \r)

  \\

   \nonumber
   &=\int_0^h        \pp\l(  \cal{O}_{s,X^{(0,g(h))}}     ;    \Delta_1^{g(h)} \in ds   \r)   \pp\l(  S_{\Delta_1^{g(h)}} \geq K g(h)  \r)

  \\

   \nonumber
       &=     \f{  \ov(Kg(h))  }{    \ov(g(h))  }   \int_0^h        \pp\l(  \cal{O}_{s,X^{(0,g(h))}}     ;    \Delta_1^{g(h)} \in ds   \r)

  \\

   \nonumber
       &=       \f{  \ov(Kg(h))  }{    \ov(g(h))  }   \int_0^h        \pp\l(     \cal{O}_{h}     ;    \Delta_1^{g(h)}    \in    ds      \r)

     =       \f{  \ov(Kg(h))  }{    \ov(g(h))  }  \pp\l(     \cal{O}_{h}     ;    \Delta_1^{g(h)} \leq h    \r)  .

\intertext{By $\l(\ref{123}\r)$ and Lemma \mbox{\ref{lemma1}},    $ \pp(  \cal{O}_{h}     ;    \Delta_1^{g(h)}     \leq   h )     \sim   \pp(\cal{O}_h)$ as $h  \to  \infty$, so~as~$h  \to  \infty$,      }

\label{contr2}
       \pp\l(   X_h \geq K g(h) ;   \cal{O}_h    \r)
        &\geq      \f{  \ov(Kg(h))  }{    \ov(g(h))  }  \pp\l(  \cal{O}_{h}      \r)
   \sim K^{-\alpha}  \pp(\cal{O}_h),
      \end{align}
   as $\ov$ is \mbox{\hyperref[rv]{regularly varying}} at $\infty$, so    $\l(\ref{finall}\r)$  contradicts $\l(\ref{contr2}\r)$, and   $\liminf_{h\to\infty}     \pp\big(    \Delta_1^{g(h)} \newline \leq~h  ;     X_{\Delta_1^{g(h)}  -}       <  g(h)   ; \cal{O}_h    \big)   / \pp(\cal{O}_h )   >  0  $. By   $\l(\ref{lowerbound}\r)$, uniformly in $y  >  K$~as~$h  \to  \infty$,
   \[
 \pp(  X_h \in g(h) dy  ;    \cal{O}_h)  \gtrsim                y^{-1-\alpha}      \f{ L(g(h)y)}{L(g(h))  }  \pp\l(  \cal{O}_h  \r)    dy      ,
    \]

  as required for the lower bound in  $\l(\ref{l6statement}\r)$, so the proof of Lemma \mbox{\ref{l6}}~is~complete.

    \e{proof}

\section{Proofs of Auxiliary Lemmas} \label{lemmasproofs}

  \b{proof}[Proof of Lemma  \hbox{\ref{lemma4}}]    Recall that  $\ov(x) = x^{-\alpha} L(x)$  for $L$  \mbox{\hyperref[sv]{slowly varying}}  at $\infty$, so as  $\ov$ is non-increasing, for~large~$N>0$,
    \b{align}
\label{lll}
 &\  \ \ \   \int_{f(Ay)}^\infty ( \ov(g(s+h) - y) - \ov(g(s))) ds
  \\ \nonumber
  &\leq        \int_{f(Ay)}^\infty ( \ov(g(s) - y) - \ov(g(s))) ds

 =  \int_{f(Ay)}^\infty    \l(     \f{  L(g(s) - y)   }{    (g(s) - y)^\alpha }    -     \f{  L(g(s) )   }{    g(s)^\alpha }   \r)   ds
\\
\nonumber
 &= \int_{f(Ay)}^\infty     \f{  L(g(s) - y)  }{    (g(s) - y)^\alpha }        \l( 1     -      \l(\f{g(s)-y}{g(s)}\r)^\alpha  \f{  L(g(s) )  }{    L(g(s) - y) }      \r)      ds
 \\
\nonumber
 &= \int_{f(Ay)}^\infty     \f{  L(g(s) - y)  }{    (g(s) - y)^\alpha }        \l( 1     -      \l(\f{g(s)-y}{g(s)}\r)^{\alpha+N}  \f{g(s)^N      L(g(s) )     }{(g(s)-y)^N    L(g(s) - y)   }      \r)      ds.

\intertext{Now, $A>B-1$, and  $x^NL(x)$ is non-decreasing in $x$ for $x>B$ in case (\mbox{\hyperref[case1]{i}}),~so  }

\nonumber
  \l(\ref{lll}\r) &  \leq
\int_{f(Ay)}^\infty     \f{  L(g(s) - y)  }{    (g(s) - y)^\alpha }        \l( 1     -      \l(\f{g(s)-y}{g(s)}     \r)^{\alpha+N}   \r)   ds.

\intertext{One can verify   $1-(1-y/g(s))^{\alpha+N} \lesssim y/g(s)$, uniformly in $y  >  0,s  >  f(Ay)$,~so}

\nonumber
    \l(\ref{lll}\r) &\lesssim      \int_{f(Ay)}^\infty     \f{  L(g(s) - y)  }{    (g(s) - y)^\alpha }   \f{y}{g(s)}       ds .

    \intertext{As $g(s)  -  y\hspace{-2pt}\geq \hspace{-2pt}(1  -  A^{-1}) g(s) $ for   $s  >  f(Ay)$, and $\ov(x)\hspace{-3pt} =\hspace{-3pt} x^{-\alpha} L(x)$~is~non-increasing,
    }

  \l(\ref{lll}\r)      &\lesssim    y \int_{f(Ay)}^\infty     \f{  L((1-A^{-1})g(s) )  }{    g(s)^{1+\alpha }}    ds.

\intertext{Applying the   uniform convergence theorem \cite[Theorem 1.2.1]{bgt89} to the slowly varying function $L$, substituting $u=g(s)$, as  $uf'(u)\ov(u)$ is decreasing, we conclude that uniformly in $y>0$ (and so also uniformly in $h>0,y>g(h)$),}

\nonumber
    \l(\ref{lll}\r) &\lesssim
  y \int_{f(Ay)}^\infty     \f{  L(g(s) )  }{    g(s)^{1+\alpha }}    ds
  =   y \int_{Ay}^\infty    \f{L(u)}{u^{1+\alpha}}   f'(u)du
   =   y \int_{Ay}^\infty    u^{-2} u  f'(u)  \ov(u)    du
\\
\nonumber
 & \leq  A  y^2 f'(Ay) \ov(Ay) \int _{Ay}^\infty    u^{-2}   du =\f{1}{2} y f'(Ay) \ov(Ay) \lesssim y f'(y) \ov(y)  .
  \end{align}

  \e{proof}

  \b{proof}[Proof of Lemma \mbox{\ref{lowerboundforrho}}]  Recall from $\l(\ref{rho}\r)$ and $\l(\ref{123}\r)$ that
  \b{align*}
  \rho(t) &=       \f{1}{\Phi(t)}    \l[   \pp( \oo_{t}  ;  \Delta_1^{g(t)} >t   )     - \ov(g(t))^2   \int_0^t      \int_0^s       \mathbb{P}   (\cal{O}_{v})   e^{-\ov(g(t))v}   dv
ds          \r].

  \intertext{If  $\rho(t)\leq 0$, then   as for all $t>1$, $\Phi(t)\geq \Phi(1)=$ constant $>0$, for all $t>1$,}

  |\rho(t)|    &\leq     \f{    \ov(g(t))^2    }{\Phi(t)}    \int_0^t      \int_0^s       \mathbb{P}   (\cal{O}_{v})   e^{-\ov(g(t))v}   dv
ds

\\

&\leq       \f{t    \ov(g(t))^2 }{\Phi(t)}    \int_0^t           \mathbb{P}   (\cal{O}_{v})   e^{-\ov(g(t))v}   dv

  \leq  \f{t^2    \ov(g(t))^2 }{\Phi(t)}
  \lesssim    t^2    \ov(g(t))^2.
  \end{align*}
Now, $\lim_{t\to\infty}  t \ov(g(t)) =0$ by $\l(\ref{condt}\r)$, and   we conclude that $\liminf_{t\to\infty} \rho(t) \geq0$.

  \end{proof}

  \b{proof}[Proof of Lemma \mbox{\ref{qh}}]
   First recall that by Theorem \mbox{\ref{recurrentthm}},
 \[
q_h(y)     \hspace{-0.1cm}  =     \hspace{-0.1cm} \f{    \Phi_y^h(t_0(y))}{    \Phi(1)}      \hspace{-0.1cm} \lim_{t\to\infty}
      \exp\l(    \int_{  t_0(y)}^{t}       \hspace{-0.1cm}    \l(  \ov(g_{y}^{h}(s) )    \hspace{-0.1cm}  +    \hspace{-0.1cm}    \rho_y^h(s)   \r)   ds
 -    \hspace{-0.1cm}   \int_1^{ t   }    \hspace{-0.1cm}   \l(  \ov(g(s))  \hspace{-0.1cm}  +  \hspace{-0.1cm}  \rho(s)   \r) ds  \hspace{-0.05cm}     \r)  \hspace{-0.1cm}  .
 \]
  Now,     by $\l(\ref{1}\r)$ in Lemma \mbox{\ref{lemma2}},  uniformly in $h>0,y>g(h)$,
   \[
   \l|  \int_{t_0(y) }^{  \infty  }   \rho_y^h(s) ds   \r|   \lesssim    \int_{t_0(y)}^\infty     \f{1}{   s \log(s)^{1+\eps} }     \l(  1 + \f{1}{ f(y) -h}  \r)  ds   ,
  \]
  and $ 1/ (f(y) -h) \leq 1/f(\d)<\infty$ since $y\geq g(h+f(\d))$,   so the $\rho_y^h$ integral is  bounded uniformly  in $h>0,y>g(h+f(\d))$.
   By  Remark \mbox{\ref{yh0}},  $\int_1^\infty \rho(s)ds  <  \infty$.

       For   $y>g(h+f(\d))>\d$,   $f(Ay)>f(A\d)$, so taking $A$ arbitrarily large if necessary,  $t_0(y) := f(Ay) \vee f(1  +  2/A)   =   f(Ay)$, then by Lemma \mbox{\ref{lemma4}}, $ \limsup_{t\to\infty}  \int_{f(Ay)}^t    ( \ov(g_{y}^{h}(s))    -   \ov(g(s))) ds   <  \infty$,~so
  \b{equation*}
   q_h(y)  \lesssim     \Phi_y^h(f(Ay))    \exp\l(       -       \int_{1}^{f(Ay)} \ov(g(s))   ds \r).
  \end{equation*}
  For the converse inequality, as $\ov$ is non-increasing,  for   $y  >  g(h)$ (so~$f(y)  >  h$),
  \begin{align*}
         &  \phantom{\leq} \  \int_{f(Ay)}^t ( \ov(g(s)) - \ov(g(s+h)-y)) ds

        &&\hspace{-15pt}\leq      \int_{f(Ay)}^t ( \ov(g(s)) - \ov(g(s+h))) ds
        \\
         &=      \int_{f(Ay)}^t  \ov(g(s)) ds      -        \int_{f(Ay)+h}^{t+h}  \ov(g(s)) ds

         &&\hspace{-15pt}\leq      \int_{f(Ay)}^{f(Ay)+h}   \ov(g(s)) ds
         \\
         &\ &&\hspace{-15pt} \leq       h \ov( g( f(Ay))) \leq      h \ov( y)      \leq f(y) \ov(y).
  \end{align*}
   Then as $y>g(h+f(\d))>\d$ and $\lim_{y\to\infty} f(y) \ov(y) =0$ by $\l(\ref{condt}\r)$,   we  conclude

  \b{equation*}
   q_h(y)  \asymp     \Phi_y^h(f(Ay))    \exp\l(       -       \int_{1}^{f(Ay)} \ov(g(s))   ds \r).
  \end{equation*}

  \begin{comment}  \n  By Lemma \mbox{\ref{lemma4}}, $  \int_{t_0(y)}^\infty \l( \ov(g_{y}^{h}(s)) - \ov(g(s))\r) ds \leq    y f'(y) \ov(y) $, and $  y f'(y) \ov(y)$ decreases to zero as $y\to\infty$  by Definition \mbox{\ref{case1}}, so  it follows that
  \b{equation*}
   q_h(y)  \asymp     \Phi_y^h(t_0(y))    \exp\l(       -       \int_{1}^{t_0(y)} \ov(g(s))   ds \r).
  \end{equation*}
   Then as $A>3$, $t_0(y) := f(Ay) \vee f(1+2/A) = f(Ay)$ for all  $y\geq1 $,  and thus
  \b{equation*}
   q_h(y)  \asymp     \Phi_y^h(f(Ay))    \exp\l(       -       \int_{1}^{f(Ay)} \ov(g(s))   ds \r).
  \end{equation*}     \end{comment}
  \end{proof}

   \b{proof}[Proof of Lemma \ref{bigjump}]   In case (\hbox{\hyperref[case1a]{ia}}), with $\Pi(dx)=u(x)dx$,

  one can easily verify that    for each  $\delta>0$,   for all large enough $x$, $u(x) \geq   x^{-1-\alpha-\delta}$ and $u(x) \leq x^{-1-\alpha+\delta}$. It follows that $u(x)$  has bounded decrease and bounded increase, and as $\ov$ is \mbox{\hyperref[rv]{regularly varying}} at $\infty$ with index $-\alpha\in(-1,0)$ in case (\hbox{\hyperref[case1]{i}}), it follows that $\ov$ has positive increase and bounded increase (see \cite[p71]{bgt89} for precise definitions of bounded decrease, bounded increase, and positive increase). Thus we can apply  \cite[Prop 2.2.1]{bgt89}, yielding that $xu(x) \asymp \ov(x)$ for all sufficiently large $x$, so
   \[
        \f{ \Pi(g(h) dv)   }{  \ov(g(h))   }    =    \f{ u(g(h)v)     g(h)dv  }{   \ov(g(h))  }   \asymp  \f{  \ov(g(h)v) g(h)   dv}{g(h)v \ov(g(h))    }   =   v^{-1-\alpha}  \f{L(g(h)v)}{L(g(h))}  dv.
   \]
   \end{proof}

  \b{proof}[Proof of Lemma \mbox{\ref{1fyh}}]  For  $t  \geq   f(Ay)$,   $A  >  3\vee  (B-1)$, as $f$ is increasing,
\[
t
 \geq f(Ay) \geq f(y) \geq f(y)-h .
\]
For $y>0$,       $h>g(y)$, and $s\leq  f(y)-h$, we have $g_{y}^{h}(s) \leq 0$, so
\[
\pp\Big( \cal{O}_s^{g_{y}^{h}}  \Big) =   \pp\l(   X_u \geq g_{y}^{h}(u),   \forall u\leq s  \r)\geq  \pp\l(   X_u \geq 0, \forall u\leq s  \r) =1,
\]  and  we conclude, as required, that
\[
\Phi_y^h(t)  =   \int_0^t  \pp\Big( \cal{O}_s^{g_{y}^{h}}  \Big) ds    \geq   \int_0^{ f(y)- h}   \pp\Big( \cal{O}_s^{g_{y}^{h}}  \Big) ds =  f(y)- h.
  \]
  \e{proof}

   \b{proof}[Proof of Lemma  \hbox{\ref{keylemma}}] By Markov's inequality, with $\lambda = \log(1/H(t)) /  B(t)$,
                        \begin{align*}

                         & \   \pp\l( X_t^{ (0,A(t))}   > B(t) \r)    &&\hspace{-6.7cm}=  \pp\l(   e^{\lambda X_t^{(0,A(t))}}   \geq  e^{\lambda B(t)}      \r)
                        \\
                             &\leq   \mathbb{E}\l[    e^{\lambda X_t^{(0,A(t))}}   \r]  e^{-\lambda B(t)} &&\hspace{-6.7cm}=   \exp\l(  t \int_0^{A(t)}  \lambda e^{\lambda x} \ov(x) dx  \r)   H(t)
                        \\
                        &\leq \exp \l(   t \f{\log(1/H(t))}{ B(t)}    e^{ \lambda A(t)} \int_0^{A(t)}  \ov(x) dx         \r)    H(t)
                        \\
                        &=  \exp\l(   t \f{\log(1/H(t))}{ B(t)}    H(t)^{ - \f{A(t)}{B(t)}} \int_0^{A(t)}  \ov(x) dx         \r)    H(t).

                        \end{align*}
                       Now, by \cite[Theorem 2.6.1(b)]{bgt89}, which applies as $\ov$ has  \mbox{\hyperref[matus]{lower  index}} $  \beta(\ov)>-1$ in cases (\hbox{\hyperref[case1]{i}}) and (\hbox{\hyperref[case2]{ii}}), there exists  $C>0$    such that for all   $A(t)>1$,

                           \[
                         \int_0^{A(t)}  \ov(x) dx  \leq  C  \ov(A(t))A(t),
                      \]
                      from which the inequality    (\ref{keyeqn}) follows immediately, as required.
                        \e{proof}

   \b{proof}[Proof of Lemma\hspace{1.5pt}\hbox{\ref{1.1}}]  By   $\l(\ref{t0y}\r)$, $t  >   t_0(y)  \geq   f(Ay).$  As  $g  =  f^{-1}$~is~increasing,
  \b{align*}
  g_{y}^{h}(t)

   =     \l(    \f{g(t+h)}{g(t)}   - \f{y}{g(t)}     \r)g(t)

   \geq      \l(    1  - \f{y}{g(t)}     \r)g(t)

   \geq      \l(    1  - \f{y}{g(f(Ay))}     \r)g(t)

   =     (    1  - A^{-1}    )g(t).
  \end{align*}
\e{proof}

      \begin{comment}
   \p{Lower Bound on Density (a)} For all $p>\alpha$, observe that $\int_y^\infty    \Pi(dx) \geq   y^p \ov(y) \to\infty$ as $y\to\infty$, and hence for all $K>0$, $\int_K^\infty  x^p     \Pi(dx) =\infty$ for all $p>\alpha$.

   \n Now, if we assume for a contradiction that $\limsup_{x\to\infty}     u(x) x^a<\infty$, where $a>p+1>\alpha+1$, then for some large constant $C$,
   \[
   \int_K^\infty       x^p \Pi(dx) =      \int_K^\infty       x^p  u(x) dx \leq   \int_K^\infty     x^{p-a} dx <\infty,
   \]
   which contradicts the fact that  $\int_K^\infty  x^p     \Pi(dx) =\infty$  for all $K>0$.

    \p{Upper Bound on Density (b)} For all $q<\alpha$, applying Potter's theorem  \cite[Theorem 1.5.6]{bgt89}, for arbitrarily small $\delta$, say $2\delta <  \alpha-q$,
    \b{align*}
    \int_y^\infty x^q    \Pi(dx)      &=   \int_0^y  \ov(y)     \f{v^{q-1} }{q-1}     dv   +   \int_y^\infty     \ov(v)     \f{v^{q-1} }{q-1}     dv
\\
    &=  y^{q}  \ov(y)   +   \int_y^\infty     \ov(v)     \f{v^{q-1} }{q-1}     dv
    \\
    &\lesssim  y^{q}  \ov(y)   +   \int_y^\infty     v^{ -\alpha +q-1 + \delta}     dv
   \\
    &\leq      y^{q}  \ov(y)   +   \int_y^\infty     v^{  -1 - \delta}     dv  <\infty.
    \end{align*}

  \n   Noting that $\lim_{y\to\infty}  y^q \ov(y) =0$, it follows that    $\int_y^\infty x^q   \Pi(dx)  \to0$ as $y\to\infty$, and hence for all $K'>0$ and $q<\alpha$, $\int_{K'}^\infty  x^q     \Pi(dx) <\infty$.

   \n Now, if we assume for a contradiction that $\limsup_{x\to\infty}     u(x) x^b>0 $, where $b<q+1<\alpha+1$, then for some large constant $K'$,
   \[
   \int_{K'}^\infty       x^q \Pi(dx) =      \int_{K'}^\infty       x^q  u(x) dx \geq   \int_{K'}^\infty     x^{q-b} dx =\infty,
   \]
   which contradicts the fact that  $\int_{K'}^\infty  x^q     \Pi(dx) <\infty$  for all ${K'}>0$.

   \n So we have shown that for all  $\delta>0$ and for all large $x$,   $u(x) \geq   x^{-1-\alpha-\delta}=x^A$ and $u(x) \leq x^{-1-\alpha+\delta}= x^B$, and thus $u(x)$  has bounded decrease and bounded increase. This important so that we can apply Potter's theorem for BI/BD (see \cite[Prop 2.2.1]{bgt89} for a precise statement).

    \p{Replicating the O-Monotone     Density Theorem} Now we are ready to prove that $x u(x) \asymp \ov(x) $, and do so by proving inequalities in each direction. Firstly, for $\lambda>1$,
    \[
     xu(x) \gtrsim    x u(x) \int_{1/\lambda}^1   y^A dy = u(x) \int_{x/\lambda}^x   \f{v^A}{x^A} dv.
    \]
    Then we can apply Potter's theorem for BI/BD to get   $\f{ u(x)  }{  u(v)  }  \gtrsim    \f{x^A}{v^A}$ for all large $x$ and $v\in[x/\lambda,x]$. Thus
     \[
       xu(x) \gtrsim      \int_{x/\lambda}^x   u(v) dv  =  \ov\l(\f{x}{\lambda}\r)  - \ov(x) \sim (\lambda^\alpha -1 ) \ov(x),
     \]
   as required for the lower bound. Similarly for the upper bound,  one can verify
   \[
    xu(x) \lesssim    x u(x) \int_{1}^\lambda   y^B dy = u(x) \int_{x}^{\lambda x}   \f{v^B}{x^B} dv
   \]
   \[
   \lesssim      \int_{x}^{\lambda x}  u(v) dv  = \ov(x)- \ov(\lambda x)   \sim (1- \lambda^{-\alpha})\ov(x),
   \]
   and so we conclude that $x u(x) \asymp \ov(x)$.

   \end{comment}

\begin{comment}

\section{Extensions} \label{extensions}   In case (\hbox{\hyperref[case1]{i}}),   the boundary at which the conditioned process is transient/recurrent is found in Corollary  \hbox{\ref{transientcorollary}}, but we only cover a small class of functions for which the conditioned process is recurrent.
 In cases (\hbox{\hyperref[case1]{i}}) and (\hbox{\hyperref[case2]{ii}}), the tail function $\ov$ satisfies $\lim_{t\to\infty} t\ov(g(t)) = 0$, but  Proposition  \hbox{\ref{ODEprop}}  applies in the case where $\lim_{t\to\infty} t\ov(g(t)) = \infty$. This lies on the recurrent side of the boundary between $I(f)<\infty$ and $I(f)=\infty$, so   the   Markov process is      recurrent under the law $\mathbb{Q}$.

The equation  $\l(\ref{ODE}\r)$ is much harder to solve as an ODE  than the corresponding equation $\l(\ref{ODE1}\r)$, but solving $\l(\ref{ODE}\r)$ should lead to a  broad set of results for the recurrent case. Moreover, it can be verified that the   precise asymptotic  result \cite[Theorem 2.2]{dr15} for the density $\pp\l( X_t\in dx\r)$ can be utilised in this case, but not when $\lim_{t\to\infty} t\ov(g(t)) = 0$.

  \b{proposition} \label{ODEprop}   For a subordinator, if $\lim_{t\to\infty}    t\ov(g(t))   =   \infty$,~as~$t  \to  \infty$,
   \b{equation}   \label{ODE}
   \mathbb{P}\l(  \cal{O}_t      \r)
    \sim   \mathbb{P}\l(    \cal{O}_t ;     \Delta_1^{g(t)} \leq t     \r)
     =        \ov(g(t))   \int_0^t      \mathbb{P}(\cal{O}_s)    e^{ -   (t-s)  \ov(g(t))}   ds.    \hspace{-0.8cm}
 \e{equation}
 \e{proposition}

 \b{proof}[Proof of Proposition  \hbox{\ref{ODEprop}}] \hspace{-5pt} As $\Delta_1^{g(t)} $ is   exponentially distributed of~rate~$\ov(g(t))$,
 \b{equation*}

  \mathbb{P}\l(  \cal{O}_t      \r)    =    \mathbb{P}\l(  \cal{O}_t      ;   \Delta_1^{g(t)} \leq t \r)      +    \mathbb{P}\l(  \cal{O}_t      |   \Delta_1^{g(t)} >t \r) e^{ - t \ov(g(t))} .

 \end{equation*}
Now, $ \mathbb{P}\l(  \cal{O}_t      |   \Delta_1^{g(t)} >t \r) \leq \mathbb{P}(\cal{O}_t)$, and  $\lim_{t\to\infty}  e^{ - t \ov(g(t))} =0$, so as $t\to\infty$,     $ \mathbb{P}\l(  \cal{O}_t      \r)  \sim    \mathbb{P}\l(  \cal{O}_t      ;   \Delta_1^{g(t)} \leq t \r) $, as required for the first claim of    Proposition  \hbox{\ref{ODEprop}}.

  For the second claim, recall the definition $\l(\ref{truncoo}\r)$. Disintegrating on the value  $\Delta_1^{g(t)}$,
 \b{align*}

   \mathbb{P}\l(  \cal{O}_t      ;   \Delta_1^{g(t)} \leq t \r) &= \int_0^t   \mathbb{P}\l(  \cal{O}_t      |   \Delta_1^{g(t)} =t_1  \r)  \ov(g(t)) e^{-t_1 \ov(g(t))} dt_1
\\
 &=    \ov(g(t)) \int_0^t   \mathbb{P}\l(  \cal{O}_{t_1,X^{(0,g(t))}  } \r)  e^{-t_1 \ov(g(t))} dt_1.

 \intertext{Now, by the law of total probability, we can rewrite the above integrand as

 \[
  \mathbb{P}\l(  \cal{O}_{t_1,X^{(0,g(t))}  } \r)  e^{-t_1 \ov(g(t))}      =   \mathbb{P}(  \cal{O}_{t_1  } ; \Delta_1^{g(t)} >t_1 ) = \pp\l( \cal{O}_{t_1}  \r)  -  \mathbb{P}\l(  \cal{O}_{t_1 } ; \Delta_1^{g(t)} \leq t_1 \r).
 \]

Disintegrating  again on the value of $\Delta_1^{g(t)}$,  it follows that}

    \mathbb{P}\l(  \cal{O}_t      ;   \Delta_1^{g(t)} \leq t \r) &=   \ov(g(t)) \int_0^t    \pp\l( \cal{O}_{t_1}  \r)    dt_1       -     \ov(g(t)) \int_0^t   \mathbb{P}\l(  \cal{O}_{t_1 } ; \Delta_1^{g(t)} \leq t_1 \r)   dt_1

\intertext{
\[
 =  \ov(g(t)) \int_0^t    \pp\l( \cal{O}_{t_1}  \r)    dt_1       -     \ov(g(t))^2 \int_0^t  \int_0^{t_1}  \mathbb{P}\l(  \cal{O}_{t_2,X^{(0,g(t))} }  \r)  e^{-t_2 \ov(g(t))}  dt_1  dt_2.
 \]
 One can repeat this procedure to deduce that for all $n\in\mathbb{N}$,

 \b{equation}\label{errorterm}
   \mathbb{P}\l(  \cal{O}_t      ;   \Delta_1^{g(t)} \leq t \r) =    \sum_{k=1}^{n-1}     (-1)^{k-1}   \ov(g(t))^k     \int_0^t \int_0^{t_1}   \cdots \int_0^{t_{k-1}}   \pp(\cal{O}_{t_k}) dt_k \cdots dt_2 dt_1
   \end{equation}

 \b{equation*}
\hspace{2.55cm}  +\hspace{-0.05cm}    (\hspace{-0.05cm}  -1\hspace{-0.05cm}  )^{n-1} \ov( \hspace{-0.01cm}g(\hspace{-0.025cm}t \hspace{-0.025cm} ) \hspace{-0.025cm})^n  \hspace{-0.15cm}      \int_0^t    \hspace{-0.25cm}      \cdots \hspace{-0.15cm}   \int_0^{t_{n-1}}    \hspace{-0.45cm}     \pp(   \hspace{-0.05cm}     \cal{O}_{t_k \hspace{-0.025cm},X^{\hspace{-0.025cm}(\hspace{-0.01cm}0\hspace{-0.01cm},\hspace{-0.01cm}g\hspace{-0.01cm}(\hspace{-0.01cm}t\hspace{-0.01cm}) \hspace{-0.025cm} )}} \hspace{-0.05cm}    )     e^{\hspace{-0.05cm}  -t_n  \ov(\hspace{-0.025cm}  g(t)\hspace{-0.025cm}  ) } \hspace{-0.025cm}   dt_{\hspace{-0.025cm}n}  \hspace{-0.05cm}  \cdots \hspace{-0.05cm}     dt_{\hspace{-0.025cm}1} \hspace{-0.025cm}.  \hspace{-0.15cm}
 \e{equation*}
The last term in $(\ref{errorterm})$ tends to 0 as $n\to\infty$ since}

\lim_{n\to\infty} | \hspace{-0.05cm} \l(\ref{errorterm}\r) \hspace{-0.05cm}| &\leq \hspace{-0.1cm}  \lim_{n\to\infty}   \ov(g(t))^n  \hspace{-0.1cm}  \int_0^t   \hspace{-0.1cm} \cdots \hspace{-0.1cm} \int_0^{t_{n-1}} \hspace{-0.15cm}   dt_n   \cdots   dt_1  =   \lim_{n\to\infty}   \f{(t\ov(g(t)))^n}{n!}\hspace{-0.05cm} =\hspace{-0.05cm} 0.

\intertext{Similarly,  the sum in $(\ref{errorterm})$ is   absolutely convergent as $n\to\infty$, and so}

     \mathbb{P}\l(  \cal{O}_t      ;   \Delta_1^{g(t)} \leq t \r) &= \sum_{k=1}^{\infty}     (-1)^{k-1}   \ov(g(t))^k     \int_0^t     \cdots \int_0^{t_{k-1}}   \pp(\cal{O}_{t_k}) dt_k \cdots   dt_1
\\
 &=  \sum_{k=1}^{\infty}     (-1)^{k-1}   \ov(g(t))^k   \int_0^t   \pp(\cal{O}_s)   \f{ (t-s)^{k-1} }{   (k-1)! } ds.
\intertext{Interchanging the order of summation and integration, we conclude that}

   \mathbb{P}\l(  \cal{O}_t      ;   \Delta_1^{g(t)} \leq t \r) &=   \ov(g(t))      \int_0^t     \pp(\cal{O}_s)  \sum_{k=1}^{\infty}     (-1)^{k-1}   \ov(g(t))^{k-1}     \f{ (t-s)^{k-1} }{   (k-1)! }   ds

\\

 &=  \ov(g(t))      \int_0^t     \pp(\cal{O}_s)  e^{ - (t-s) \ov(g(t))} ds  .
    \end{align*}

 \e{proof}

 \end{comment}

\begin{comment}

\subsection{Entropic Repulsion}

  For conditioned Brownian motion in the recurrent case where $I(f)=\infty$, the conditioned process generally doesn't use all of its allowance of local time.
   \ \\
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  This phenomenon is related to entropic effects, which cause the process to stay far away from breaking the constraint to allow for more fluctuations.
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   ``Entropic repulsion'' describes situations  where the most likely way for a process to satisfy an imposed condition  is for the process to satisfy an even stronger condition.
 \ \\
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 In the case where $I(f)=\infty$, we can study decreasing functions $w$  such that $\lim_{t\to\infty}w(t)=0$, and  $\lim_{t\to\infty}     \mathbb{Q}(     L_t \leq     w(t) f(t)    ) =1$. The set of such functions is called the ``entropic repulsion envelope'' for the function $f$, and is denoted by
  \[
  R_f:=   \l\{     w:  w\downarrow 0,   \lim_{t\to\infty}     \mathbb{Q}(     L_t \leq     w(t) f(t)    ) =1    \r\}.
   \]

\n Theorem (Kolb, Savov 2016)        For 1-dimensional Brownian motion, when $I(f)=\infty$,  the repulsion envelope is given by
  \[
   w\in R_f \iff        \lim_{h\to\infty}    \int_h^{f(      w(h) g(h) )}      \f{1}{\sqrt{g(s)}} ds =0.
  \]

     In case (\mbox{\hyperref[case1]{i}}), with $I(f)=\infty$,
  \[
   w\in R_f \iff        \lim_{h\to\infty}    \int_h^{f(      w(h) g(h) )}     \ov(g(s)) ds =0.
  \]
\end{comment}

\subsection{Acknowledgements} Thanks to Mladen Savov for suggesting the topic and   discussing the work at various stages.  Funding was~provided~by~EPSRC.

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