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\newtheorem{defin}{Definition}

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\title{The local asymptotic estimation for the supremum of a random walk with generalized strong subexponential summands
\thanks{Research supported by the National Natural Science Foundation of China
(No.11071182 ), the National Natural Science Foundation of China
(No.11401415), the Natural Science Foundation of the Jiangsu Higher
Education Institutions of China (No.13KJB110025).}}
\author{\small
Yuebao Wang$^{1}$\thanks{Corresponding author.
Telephone: 86 512 67422726. Fax: 86 512 65112637. E-mail:
ybwang@suda.edu.cn}~~ Hui Xu$^{1}$~~  Dongya Cheng$^{1}$~~ and~~ Changjun Yu$^{2}$
\\
{\footnotesize\it 1. School of Mathematical Sciences, Soochow
University, Suzhou 215006, China}\\ {\footnotesize\it 2. School of
Sciences, Nantong University, Nantong 226019, China}}
\date{}

\maketitle {\noindent\small {\bf Abstract }}\\

{\small In this paper, the local asymptotic estimation for the supremum of a random walk and its applications are presented. The summands of the random walk have common long-tailed and generalized strong subexponential distribution. This distribution class and the corresponding generalized local subexponential distribution class are two new distribution classes with some good properties. Further, some long-tailed distributions with intuitive and concrete forms are found, which show that the intersection of the two above-mentioned distribution classes with long-tailed distribution class properly contain the strong subexponential distribution class and the locally subexponential distribution class, respectively.\\

{\it Keywords:} random walk; supremum; local asymptotic estimation;
generalized strong subexponential distribution; generalized locally subexponential distribution} \\

{\it 2000 Mathematics Subject Classification:} Primary 60E05;  Secondary 60F99\\

\section{Introduction}

In this paper, we primarily study the local asymptotics
for the supremum of a random walk generated by summands with
common long-tailed and generalized strong subexponential distribution.
The generalized strong subexponential distribution class and
corresponding generalized local subexponential distribution class
are two new distribution classes with some good properties.
Therefore, we introduce some related concepts and notations in this section.

Unless otherwise stated, we always assume that a
random variable (r.v.) $X$ has a proper
distribution $F$ supported on $[0,\infty)$, that is its tail distribution function
$\overline{F}(x)=P(X>x), x\in (-\infty,\infty)$ is always positive. By definition, a
distribution $F$ is said to be heavy-tailed, if for all $\alpha>0$,
\begin{eqnarray*}
\int_{0}^{\infty}e^{\alpha y}dF(y)=\infty.
\end{eqnarray*}
Otherwise, $F$ is called light-tailed. As is known to all,
heavy-tailed distributions have important applications in various
fields of applied probability, such as risk
theory, queuing system, warehousing management, branching theory,
communication net and infinite divisible distribution theory, see,
for example, Embrechts et al. (1997) and Foss et al. (2013).
Further, some common heavy tailed distributions are also studied and applied
in the statistics, see Zeller et al. (2012), Tavangar and Hashemi (2013), Sultan and Al-Moisheer (2013),
and so on. So they attract much interest of the researchers. However, the
heavy-tailed distribution class is too large and it contains some
distributions which can not be ``dominated'', so some subclasses of
the heavy-tailed distribution class with good properties were
introduced. Here we first recall some existing subclasses of the
heavy-tailed distribution class, and then introduce some new ones.

We say that a distribution $F$ belongs to the subexponential distribution class, denoted
by $F\in{\cal{S}}$, if
$$\overline{F^{*2}}(x)\sim 2\overline{F}(x),$$
that is $\lim\overline{F^{*2}}(x)(\overline{F}(x))^{-1}=2$, here and after, all limits refer to $x$ tending to infinity, unless otherwise stated. The
subexponential distribution class was introduced by Chistyakov
(1964) in the study of the branching process, where it was proved
that the subexponential distribution class is contained in the following
heavy-tailed distribution subclass. We say that a distribution $F$ belongs to
the long-tailed distribution class, denoted by $F\in{\cal{L}}$, if
for any $y\in(-\infty,\infty)$,
$$\overline{F}(x+y)\sim\overline{F}(x). $$

For convenience, we write
\begin{eqnarray*}
C_*(F)=\liminf\overline{F^{*2}}(x)(\overline{F}(x))^{-1}{\rm{~and~}}C^*(F)
=\limsup\overline{F^{*2}}(x)(\overline{F}(x))^{-1}.
\end{eqnarray*}
It was obtained by Theorem 1 of Foss and Korshunov (2007) that for any
heavy-tailed distribution $F$, $C_*(F)=2$. And it is obvious that
$F\in{\cal{S}}$ {if and only if} $C_*(F)=C^*(F)=2$, which means that
for a subexponential distribution $F$, compared with other
distributions, the fluctuations of the ratios
$\overline{F^{*2}}(x)\big(\overline{F}(x)\big)^{-1}$ is minimal as
$x\to\infty$. So to some extent, we may regard  such distribution
$F$ as ``optimal". Also, we may say a distribution $F$ is
``controllable" if $$2\le C^*(F)<\infty.$$ And when $C^*(F)=\infty$, we
say that the distribution $F$ is ``uncontrollable".

In a probabilistic model with heavy-tailed distributions, if we may
choose distributions freely, then the first choice is of course the
subexponential distributions. However, due to the complexity of the real world, the
distributions usually are not decided by us. So it is necessary to study
``controllable" distributions, even ``uncontrollable" distributions.

Kl\"{u}ppelberg (1990) first considered the ``controllable" distributions and called them ``weak idempotents". Shimura and
Watanabe (2005) called the distributions
generalized subexponential and denoted the class of such
distributions by $\cal{OS}$. In the terminology of Bingham et al.
(1987), the distributions from the class $\cal{OS}$ are called
O-regularly varying functions. Here we continue to use the notation
$\cal{OS}$.

The distribution class $\cal{OS}$ is a rather large class {which} contains many
heavy-tailed and light-tailed distributions. For research on $\cal{OS}$,
besides the above-mentioned literature, the
reader can refer to Kl\"{u}ppelberg and Villasenor (1991), Watanabe and
Yamamura (2010), Lin and Wang (2012), Cheng and Wang (2012), Cheng
et al. (2012), Yu and Wang (2013), Gao et al. (2013),
Beck et al. (2013), Xu et al. (2015a,b,c), and so on.

In present paper, we are interested in some subclasses of the class
${\cal{OS}}$, which respectively correspond two subclasses of the class
${\cal{S}}$ as follows.

We say that a distribution $F$ belongs to the strong subexponential distribution class,
denoted by $F\in{\cal{S}}^*$, if $0<EX<\infty$ and
$$\int_{0}^{x}\overline{F}(x-y)\overline{F}(y)dy\sim 2
EX\overline{F}(x).$$
In the above formulas,
if the distribution $F$ is supported on $(-\infty,\infty)$, then
$EX$ is replaced by $EX^{+}$, where $X^{+}=X\textbf{1}(X\ge0)$.

We say that a distribution $F$ belongs to the locally long-tailed distribution class, denoted by
$F\in{\cal{L}}_{\Delta_T}$, {where $T$ is some positive constant or
$\infty$}, if for some constant $x_0>0$, $F(x+\Delta_T)=P(X\in
{x+\Delta_T})>0$ for all $x\ge x_0$ and the relationship
$$F(x+y+\Delta_T)\sim F(x+\Delta_T)$$ holds uniformly for all
$y\in(0,T]$, where $\Delta_T=(0,T],~x+\Delta_T=(x,x+T]$ when
$T<\infty$, and $\Delta_T=(0,\infty),~x+\Delta_T=(x,\infty)$
when $T=\infty$. Further, we say that a distribution $F$ belongs to the locally subexponential distribution class, denoted by $F\in{\cal{S}}_{\Delta_T}$, if for some
$0<T\le \infty$, $F\in{\cal{L}}_{\Delta_T}$ and
$$F^{*2}(x+\Delta_T)\sim 2F(x+\Delta_T).$$

The strong subexponential distribution and the
locally subexponential distribution were introduced by
Kl\"{u}ppelberg (1988) and Asmussen et al. (2003), respectively

. Inspired by the distribution classes $\cal{S}^*$, ${\cal{S}}_{\Delta_T}$ and
$\cal{OS}$, we introduce the following two new distribution
classes which are the main object of study of present paper.

\begin{defin}
We say that a distribution $F$ belongs to the generalized strong subexponential distribution class, denoted by $F\in{\cal OS}^*$, if
$$C^\otimes(F)=\limsup\int_{0}^{x}\overline{F}(x-y)\overline{F}(y)dy\big(\overline {F}(x)\big)^{-1}<\infty.$$
\end{defin}

\begin{defin}
We say that a distribution $F$ belongs to the generalized locally subexponential distribution class for some
$0<T\le \infty$, denoted by $F\in{\cal OS}_{\Delta_T}$, if
for some constant $x_0>0$, $F(x+\Delta_T)>0$ for all $x\ge x_0$ and
$$C^T(F)=\limsup F^{*2}(x+\Delta_T)\big(F(x+\Delta_T)\big)^{-1}<\infty.$$
\end{defin}

Obviously, like the class ${\cal{OS}}$, the classes ${\cal{OS}}^*$ and
${\cal{OS}}_{\Delta_T}$ contain many heavy-tailed distributions
and light-tailed distributions. Moreover, the classes ${\cal{OS}}^*$
and ${\cal{OS}}_{\Delta_T}$ have also certain ``controllability''.

Existing research and application on class $\mathcal{OS}^*$, for example, can be found in Proposition 1.1, 1.2, 1.4 and Lemma 2.1 of Xu et al. (2015b).

Similar to $C_*(F)$, we write
\begin{eqnarray*}
C_\otimes(F)=\liminf\int_0^x\overline{F}(x-y)\overline{F}(y)dy(\overline{F}(x))^{-1}
\end{eqnarray*}
and
\begin{eqnarray*}
C_T(F)=\liminf F^{2*}(x+\d)(F(x+\d))^{-1}.
\end{eqnarray*}

For a heavy-tailed distribution $F$, apart from proving the fact
that $C_*(F)=2$, Foss and Korshunov (2007) also proved  that
$C_{\otimes}(F)=2EX$. However, for a light-tailed distribution $F$,
the equalities $C_{\otimes}(F)=2EX$ do not necessarily hold,
see Foss and Korshunov (2007). Similarly, for a locally heavy-tailed distribution $F$,
the equality $C_{T}(F)=2$ for some $0<T<\infty$ also does not necessarily hold, but if $F\in{\cal{L}}_{\Delta_T}$ for some $0<T<\infty$, then $C_{T}(F)=2$, see Proposition 4.1 and Remark 4.1 of Chen et al. (2013).

The remainder of this paper consists of four sections. In Section 2,
the relationships among the two new distribution classes and some
existing related ones are discussed. Some examples of long-tailed distribution show that
the class $\mathcal{OS}^*$ and
the class $\mathcal {OS}_{\Delta_T}$ properly contain the
class $\mathcal{S}^*$ and the class $\mathcal {S}_{\Delta_T}$, respectively. It should be said that the methods of construction of these distributions are not trivial, but these distributions are not particularly weird, especially their integral tail distributions are much more normal. And proofs of this examples are given in the Section 5. In Section 3, the local asymptotic estimation for the supremum of a
random walk is presented, where the summands of the random walk have
common long-tailed and generalized strong subexponential distribution. To this end, we find out some relationship between the random walk with heavy-tailed summands and the random walk with light-tailed summands. Some applications of the above results are given in Section 4.

\section{The relationships among the distribution classes}
\setcounter{equation}{0}
\setcounter{thm}{0}\setcounter{Corol}{0}\setcounter{lemma}{0}

\subsection{ The relation between the classes
$\cal{L}\cap{\cal{OS}^*}$ and $\cal{S}^*$.}

\begin{pron}\label{p201}
The inclusion relation $\cal{S}^*\subset\cal{L}\cap{\cal{OS}^*}$ is
proper.
\end{pron}

\proof Obviously, the inclusion relation $\cal{S}^*\subset\cal{L}\cap{\cal{OS}^*}$ holds. So, we just prove that the relationship is proper through the following two types of distributions.

\begin{exam}\label{e201}
Let $m\geq 1$ be any integer. Choose any constant $\alpha\in (m^{-1},
1+m^{-1})$ and any constant $x_1>4^{m\alpha(m\alpha-1)^{-1}}$. For all
integers $n\geq1$, let $x_{n+1}=x_n^{2-(m\alpha)^{-1}}$. Clearly,
$x_{n+1}>4x_n$ and $x_n\to\infty$ as $n\to\infty$. Now, define the
distribution $F$ as follows:

\begin{eqnarray}
\overline{F}(x)&=&\textbf{\emph{1}}(x< 0)+(x_1^{-1}(x_1^{-\alpha}-1)x+1)
\textbf{\emph{1}}(0\leq x< x_1)\nonumber\\
&&\ \ \ \ \ \ \ \ \ +\sum\limits_{n=1}^{\infty}
\Big(\big(x_n^{-\alpha}+(x_n^{-2\alpha-1+m^{-1}}-x_n^{-\alpha-1})(x-x_n)\big)\textbf{\emph{1}}
(x_n\leq x< 2x_n)\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +x_n^{-2\alpha+m^{-1}}\textbf{\emph{1}}(2x_n\leq x< x_{n+1})\Big),~x\in(-\infty,\infty).\label{201}
\end{eqnarray}
Further, let
$$\overline{G_m}(x)=(\overline{F}(x))^m=\overline{F}^m(x),~x\in(-\infty,\infty).$$
Then $G_m\in({\cal{S}}\cap{\cal{OS^*}})\setminus{\cal{S^*}}$.
\end{exam}

\begin{exam}\label{e202}
Let $m\geq 1$ be any integer. Choose any constant $\alpha\in
(2+2m^{-1},\infty)$ and any constant $x_1>4^{\alpha}$. And, for all
integers $n\geq1$, let $x_{n+1}=x_n^{1+{\alpha}^{-1}}$. Clearly,
$x_{n+1}>4x_n$ and $x_n\to\infty$ as $n\to\infty$. Now, define the
distribution $F$ as follows:
\begin{eqnarray}
\overline{F}(x)&=&\textbf{\emph{1}}(x< 0)+(x_1^{-1}(x_1^{-\alpha}-1)x^{2^{-1}}+1)\textbf{\emph{1}}(0\leq x< x_1^2)\nonumber\\
&&\ \ \ \ \ \ \ +\sum\limits_{n=1}^{\infty}
\Big(\big(x_n^{-\alpha}+(x_n^{-\alpha-2}-x_n^{-\alpha-1})(x^{2^{-1}}-x_n)\big)
\textbf{\emph{1}}(x_n^2\leq x< 4x_n^2)\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ +x_n^{-\alpha-1}\textbf{\emph{1}}(4x_n^2\leq x< x_{n+1}^2)\Big),~x\in(-\infty,\infty).\label{202}
\end{eqnarray}
Further, let $G_m$ be the same as in  Example \ref{e201}, then
$G_m\in({\cal{L}}\cap{\cal{OS}^*})\setminus{\cal{S}}$, thus $G_m\notin
{\cal{S}^*}$.
\end{exam}

Therefore, the proposition is proved.\hfill$\Box$

\subsection{ The relation between the classes ${\cal{L}}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T}$ and ${\cal{S}}_{\Delta_T}$}

\begin{pron}\label{p202}
For all $0<T\leq \infty$, the inclusion relation
${\cal{S}}_{\Delta_T}\subset{\cal{L}}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T}\subset{\cal{L}}_{\Delta_T}\cap{\cal{OS}}$
is proper.
\end{pron}

\proof~When $T=\infty,$ the corresponding counterexamples showing
that there exist some distributions belonging to the class
${\cal{L}}\cap{\cal{OS}}\setminus{\cal{S}}$ may be found in Leslie
(1989), Lin and Wang (2012), Example \ref{e202} and Example \ref{e205} below.
So we only prove the result in the case
that $0<T<\infty.$ First, we prove a simple fact that
${\cal{L}}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T}\subset{\cal{L}}_{\Delta_T}\cap{\cal{OS}}$.
Let $V\in {\cal{L}}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T}$ for some
$0<T<\infty$, we have

\begin{eqnarray*}
\int_0^x\overline{V}(x-y)dV(y)&\le&\sum\limits_{k=0}^{[xT^{-1}]}
\sum\limits_{l=0}^{\infty}V(x+lT-(k+1)T+\Delta_T)V(kT+\Delta_T)\nonumber\\
&=&O\Big(\sum\limits_{l=0}^{\infty}\sum\limits_{k=0}^{[xT^{-1}]}
\int_{kT}^{(k+1)T}V(x+lT-y+\Delta_T)dV(y)\Big)\nonumber\\
&=&O\Big(\sum\limits_{l=0}^{\infty}\int_0^xV(x+lT-y+\Delta_T)dV(y)\Big)=O(\overline{V}(x)),
\end{eqnarray*}
thus $V\in\cal{OS}$, where $a(x)=O(b(x))$ mean that $\limsup a(x)(b(x))^{-1}\le1$ for two positive functions $a$ and $b$.

The following Example \ref{e205} shows that the distribution class
${\cal{L}}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T}$ is properly included in the distribution class ${\cal{L}}_{\Delta_T}\cap \cal{OS}$. Now, we give three counterexamples to show that the
inclusion relationship ${\cal{S}}_{\Delta_T}\subset{\cal{L}}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T}$
is also proper. To this end, we first introduce
a concept of distribution. For some distribution $F$ with a finite
and positive mean $EX$, we say that the distribution $F^{I}$ defined by
$$F^I(x)=(EX)^{-1}\int_{0}^{x}\overline{F}(y)dy\emph{\textbf{\emph{1}}}(x>0), ~~x\in(-\infty,\infty)$$
is the integrated tail distribution (or equilibrium distribution) of the distribution $F$. Related work on the integrated tail distribution can be found in Kl\"{u}ppelberg (1988), Korshunov (1997), Li and Xu (2008), and so on.

\begin{exam}\label{e203}
For any $m\geq 1$, let $G_m$ be the same as in Example \ref{e201} {or Example
\ref{e202}.} Then for all $0<T<\infty,$
$G_m^I\in({\cal{L}}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T})\setminus{\cal{S}}_{\Delta_T}.$
\end{exam}

In order to give the third counterexample, we first introduce some
relevant notions and notations. We say that a distribution $F$
belongs to the exponential distribution class with the index
$\gamma\geq 0$, denoted by $F\in{\cal L}(\gamma)$, if for all
$t\in(-\infty,\infty)$, $$\overline {F}(x+t)\sim e^{-\gamma
t}\overline {F}(x).$$ We say that a distribution $F$ belongs to the
convolution equivalent distribution class with the index $\gamma\geq
0$, denoted by $F\in{\cal S}(\gamma)$, if $F\in{\cal L}(\gamma)$,
$M_{\gamma}(F)=\int_0^{\infty}e^{\gamma y}dF(y)<\infty$ and
$$\overline{F^{*2}}(x)\sim 2M_{\gamma}(F){\overline F}(x).$$
Obviously, when $\gamma=0$, ${\cal{L}}(0)$=$\cal{L}$ and
${\cal{S}}(0)$=$\cal{S}$; when $\gamma>0$, the distributions in
${\cal L}(\gamma)$ are light-tailed. The classes $\mathcal{L}(\gamma)$ and $\mathcal{S}(\gamma)$ were introduced by Chover et al. (1973, a, b) for $\gamma>0$. Bertoin and Doney (1996) note that, however, in definitions of the class $\mathcal{L}(\gamma)$ and the class $\mathcal{S}(\gamma)$, if $\gamma>0$, and if the distribution $F$ is lattice, then $x$ and $T$ should be restricted to values of the lattice span.

Further, for a distribution $F$, if $M_{\gamma}(F)<\infty$ for
some $\gamma>0$, we may define a new distribution as follows.
\begin{eqnarray*}
F_{\gamma}(x)=(M_{\gamma}(F))^{-1}\int_0^{x}e^{\gamma
y}dF(y)\textbf{1}(x\ge0),~x\in(-\infty,\infty),
\end{eqnarray*}
which is called the $\gamma$-transform or the Escher transform of
the distribution $F$. {Similarly, we can define the
$-\gamma$-transform of a distribution $F$ for any $\gamma>0$.}

\begin{exam}\label{e204}
Kl\"{u}ppelberg and Villasenor (1991) { found} two distributions
$F_i\in{\cal S}(\gamma)$ for some $\gamma>0,~i=1,2,$ but
$F=F_1*F_2\in{\cal L}(\gamma)\setminus{\cal S}(\gamma)$. Then for
all $0<T<\infty$, $F_{\gamma}\in({\cal L}_{\Delta_T}\cap{\cal
OS}_{\Delta_T})\setminus{\cal S}_{\Delta_T}$.\hfill$\Box$
\end{exam}

\begin{re}
We point out that the distribution $F_{\gamma}\in {\cal
L}_{\Delta_T}\setminus{\cal S}_{\Delta_T}$ for all $0<T<\infty$ in
Example \ref{e204} was firstly introduced by Proposition 2.1 of Chen
et al. (2013). In addition, there Example \ref{e203} and Example \ref{e204}
give two new ways to find more distributions in the class
$(\cal{L}\cap\cal{OS})\setminus{\cal S}$ and its subclasses.
\end{re}

\subsection{ The relation between the classes  $\cal{L}\cap{\cal{OS}}$ and
$\cal{L}\cap{\cal{OS}^*}$.}

\begin{pron}
The inclusion relation $
\cal{L}\cap{\cal{OS^*}}\subset\cal{L}\cap{\cal{OS}}$ is proper.
\end{pron}
\proof First, using the method of Lemma 9 of Denisov et al. (2004), we can prove the fact that $\cal{L}\cap{\cal{OS^*}}\subset\cal{L}\cap{\cal{OS}}$.
Next, we prove that the above inclusion relation is proper by
using the following example. To this end, we recall a distribution
in the distribution class $ \cal{L}\cap{\cal{OS}}\backslash\cal{S}$,
which was found by Lin and Wang (2012).
\begin{exam}\label{e205}
Let $x_1>1$ be any given number, and let $x_{n+1}=(2x_n)^2$,
$n\geq1$. For any $\alpha\in (0,1)$, define
\begin{eqnarray}
\overline{F}(x)&=&\textbf{\emph{1}}(x< 0)+(x_1^{-1}(x_1^{-\alpha}-1)x+1)\textbf{\emph{1}}(0\leq x< x_1)\nonumber\\& &\ \ \ \ \ \ \ \ \ +\sum\limits_{n=1}^{\infty}\Big(\big(x_n^{-\alpha}+(2^{-2\alpha}x_n^{-2\alpha-1}-x_n^{-\alpha-1})(x-x_n)\big)
\textbf{\emph{1}}(x_n\leq x< 2x_n)\nonumber\\
& &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +(2x_n)^{-2\alpha}\textbf{\emph{1}}(2x_n\leq x< x_{n+1})\Big),~x\in(-\infty,\infty).\label{227}
\end{eqnarray}
For any positive integer $m\in(\alpha^{-1},2\alpha^{-1})$, let
$\overline{G_m}(x)=\overline{F}^m(x),~~x\in(-\infty,\infty).$ It is
obvious that the distribution $G_m$ has a finite mean. Lin and Wang
(2012) has proved that
$G_m\in(\cal{L}\cap{\cal{OS}})\backslash\cal{S}$. Further, we
have $G_m\notin\cal{OS}^*$, $G_m^I\in(\cal{L}\cap{\cal{OS}})\setminus{\cal{S}}$ and
$G_m^I\in{\cal{L}}_{\Delta_T}\setminus{\cal{OS}}_{\Delta_T}$ for all $0<T<\infty$.\hfill$\Box$
\end{exam}

\section{Local asymptotic estimations} \setcounter{equation}{0}
\setcounter{thm}{0}\setcounter{Corol}{0}\setcounter{lemma}{0}\setcounter{re}{0}

In this section, we try to deliver local asymptotic estimations
for the supremum of a random walk, where the distributions
of the summands of the random walk belong to the class
$\cal L\cap{\cal{OS^*}}$, or equivalently, the integrated tail
distributions of the summands belong to the class ${\cal
L}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T}$ for any $0<T<\infty$, {see
Lemma \ref{l35} below}. This explains that distributions from the
classes $\cal L\cap{\cal{OS^*}}$ {and  ${\cal
L}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T}$} possess good properties,
thus they have important value in applications.

In the following, we first introduce some concepts of a random walk
and the main result of this paper. Then  we give some lemmas in
the second subsection. The proof of the main result will be
presented at last.

\subsection{Related concepts and main result}

Let $\{X_i: i\geq 1\}$ be a sequence of independent, identically
distributed r.v.s with a common non-degenerate distribution $F$
supported on $(-\infty,\infty)$. Denote the random walk by
$\{S_n=\sum_{i=1}^{n}X_i:n\ge0\}$, where $S_0=0$,  and the supremum
of the random walk by $M=\sup_{n\ge0}S_n$ with a distribution $W$
{supported on $[0,\infty)$}. Assume that $$-\infty
<-\mu=EX_1<0,$$ then we know that $S_n$ drifts to $-\infty$ and $W$ is a proper
distribution.

Further, let $\tau_+=\inf\{n\ge1:S_n>0\}$ be the first ascending
ladder-epoch and $S_{\tau_+}$ the first ascending ladder hight with
a defective distribution $F_+$ {supported on $[0,\infty)$}, i.e.,
$0<p=F_+(\infty)<1$. Denote $G(x)=p^{-1}F_+(x),
x\in(-\infty,\infty)$, then $G$ is a proper distribution {supported
on $[0,\infty)$}. It is well known that, for any $0<T\le\infty$ and
$x\ge0$,
\be\label{31}W(x+\d)=(1-p)\sum_{n=1}^{\infty}p^nG^{*n}(x+\d),\ee
see Asmussen et al. (2002) or Asmussen et al. (2003).
\medskip

For the random walk $\{S_n:n\ge0\}$, if $F\in\cal S^*$, then
\be\label{32}W(x+\d)\sim\mu^{-1}T\overline F(x),\ee
for any $0<T<\infty$, see Asmussen et al. (2002), and so on. Further, Asmussen et al. (2003) show that $W\in{\cal S}_{\Delta_T}$ for any $0<T<\infty$. Naturally, one hopes to know that if $F\in\cal L\cap{\cal{OS^*}}$,
then how to estimate $W(x+\d)$? And, what is distribution of the supremum $M$? Our answer is as follows.

\begin{thm}\label{t31}
For the random walk $\{S_n:n\ge0\}$, if $F\in\cal L$,
then for any $0<T<\infty$,
\be\label{33}\liminf W(x+\d)(\overline F(x))^{-1}=\mu^{-1}T.\ee
Further, if $F\in\cal
L\cap{\cal{OS^*}}$ and
\be\label{34}C^\otimes(F)<\mu+2EX_{1}^{+},\ee
then for any $0<T<\infty$,
\be\label{35}
\limsup W(x+\d)\big(\overline F(x)\big)^{-1}\le\mu^{-1}T\lo(1-\mu^{-1}(C^\otimes(F)-2EX_{1}^{+})\ro)^{-1},
\ee
and $W\in{\cal L}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T}$. In
particular, if $C^\otimes(F)=2EX_{1}^{+}$, namely $F\in{\cal S}^*$,
then (\ref{32}) holds. In addition, if $F\in(\cal L\cap{\cal{OS^*}})\setminus \cal {S}^*$,
then $W\in({\cal L}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T})\backslash{\cal{S}}_{\Delta_T}$.
\end{thm}

\begin{re}
Here we note that the theorem gives also us a new way to find more distributions in
the class $({\cal L}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T})\backslash
{\cal{S}}_{\Delta_T}$ for any $0<T<\infty$.

In addition,

we give the following results to illustrate the condition (\ref{34})

\end{re}

\begin{pron}\label{p31}
There exists a long-tailed and generalized strong subexponential distribution $F$ supported on $(-\infty,\infty)$ satisfying the condition (\ref{34}).
\end{pron}

\subsection{Some lemmas}
In this section, we prepare more lemmas on
the local distributions, which will be used in the proof of Theorem
3.1 and also have their own independent value.

First, we recall a known fact. If a distribution
$V\in{\cal{L}}_{\Delta_T}$ for some $0<T\le \infty$, then
\begin{eqnarray*}
\mathcal{H}_{\Delta_T}(V)&=&\{h ~{\rm{on}}~
[0,\infty):~h(x)\uparrow\infty,~h(x)=o(x)\ \mbox{and}\\
&&V(x+t+\Delta_T)\sim V(x+\Delta_T)\ \mbox{holds uniformly for
all}~|t|\leq h(x)\}\neq\emptyset.
\end{eqnarray*}
And if $h\in\mathcal{H}_{\Delta_T}(V)$ and $h(x)\ge
h_1(x)\uparrow\infty$, then $h_1\in\mathcal{H}_{\Delta_T}(V)$ too.
Particularly, when $T=\infty$, we denote
$\mathcal{H}_{\Delta_{\infty}}(V)$ by $\mathcal{H}(V)$.

\begin{lemma}\label{l31}(i) If $V\in{\cal L}_{\d}$ for some $0<T\le \infty$, and for some $h\in\mathcal{H}_{\Delta_T}(V)$,
\be\label{36}\int_{h(x)}^{x-h(x)}V(x-y+\d)dV(y)=O(
V(x+\Delta_T)),\ee then  $V\in{\cal OS}_{\d}$.

(ii) If $V\in{\cal{L}}_{\Delta_T}\cap{\cal OS}_{\d}$, then for all
$h\in\mathcal{H}_{\Delta_T}(V)$, {(\ref{36}) holds and}
\be\label{37}\limsup\int_{h(x)}^{x-h(x)}V(x-y+\d)(V(x+\Delta_T))^{-1}dV(y)=C^T(V)-2.\ee\end{lemma}

\proof  By $V\in{\cal L}_{\d}$ and a standard method, we have
\be\label{38}V^{*2}(x+\d)\sim
2V(x+\Delta_T)+\int_{h(x)}^{x-h(x)}V(x-y+\d)dV(y),\ee thus
$V\in{\cal OS}_{\d}$ follows immediately from   (\ref{38}) and
(\ref{36}).

On the other hand, if $V\in{\cal OS}_{\d}$, {then (\ref{36}) and}
(\ref{37}) follow directly from (\ref{38}).$\hfill\Box$\medskip

\begin{lemma}\label{l32}If $V\in{\cal{L}}_{\Delta_T}\cap{\cal OS}_{\d}$ for some
$0<T\le\infty$, then for all $n\ge1$,
\be\label{39}\limsup V^{*n}(x+\d)(V(x+\Delta_T))^{-1}\le\sum_{k=0}^{n-1}(C^T(V)-1)^{n-1-k}.\ee
\end{lemma}
\proof Apparently, (\ref{39}) holds for $n=1,2$. We assume that
(\ref{39}) holds for $n=m$ and aim to show that it holds for $n=m+1$
too.

For any $n\ge 1$, denote
$$C_n^T(V)=\limsup V^{*n}(x+\d)(V(x+\Delta_T))^{-1}.$$ For any
$h\in\mathcal{H}_{\Delta_T}(V)\cap \mathcal{H}_{\Delta_T}(V^{*m})$,
by a standard method, we obtain \be\label{q10}V^{*(m+1)}(x+\d)\sim
V(x+\Delta_T)+V^{*m}(x+\d)+I(x),\ee where
\bea\label{q11}I(x)&\le&\int_{h(x)-T}^{x-h(x)+T}V^{*m}(x-y+\d)dV(y)\nonumber\\&\lesssim&
C_m^T(V)\int_{h(x)-T }^{x-h(x)+T}V(x-y+\d)dV(y).\eea It follows from
(\ref{39})-(\ref{q11}) and Lemma \ref{l31} that \beq
C_{m+1}^T(V)&\le&
C_m^T(V)(C^T(V)-1)+1\\&\le&\sum_{k=0}^{m}(C^T(V)-1)^{m-k}, \eeq
namely (\ref{39}) holds for $n=m+1$. $\hfill\Box$\medskip

\begin{lemma}\label{lem5}
~~Let $V$ be a proper distribution on $[0,\infty)$. If
$V\in{\cal{L}}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T}$ for some
$0<T\le\infty$, then for arbitrary $\varepsilon>0 $, there exists
$x_1>0$ and $K=K(\varepsilon, x_1)>0$ such that for all $x\geq x_1$
and $n\geq1$,
\begin{eqnarray}
{V^{*n}}(x+\Delta_T)\leq K(C^T(V)-1+\varepsilon)^n
{V}(x+\Delta_T).\label{tj1}
\end{eqnarray}
\end{lemma}

\proof For any $h\in\mathcal{H}_{\Delta_T}(V)$, by the condition
$V\in{\cal{L}}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T}$, we know that
for any $\varepsilon\in (0,1)$, there exists a positive number $x_1$
large enough such that when $x\ge x_1$, $h(x)>2T$,

\begin{equation}
\left|{V}(x-y+\Delta_T)({V}(x+\Delta_T))^{-1}-1\right|<8^{-1}\varepsilon
\quad{\rm{uniformly ~for}}\quad|y|\leq h(x)\label{305}
\end{equation}
and
\begin{equation}
\int_{h(x)-T}^{x-h(x)+T}{V}(x-y+\Delta_T)dV(y)<\lo(C^T(V)-
2+8^{-1}\varepsilon\ro) {V}(x+\Delta_T).\label{306}
\end{equation}
We now prove the lemma by induction. When $n=1$, (\ref{tj1}) is
obvious. Assume that (\ref{tj1}) holds for a fixed integer $n\geq1$,
we now show that it holds for $n+1$. Denote
\begin{equation*}
A_n=\sup_{x\geq
x_1}{V^{*n}}(x+\Delta_T)({V}(x+\Delta_T))^{-1},~n\geq 1.
\end{equation*}
We have
\begin{eqnarray}
V^{*(n+1)}(x+\Delta_T)&=&\int_0^{h(x)}{V^{*n}}(x-y+\Delta_T)dV(y)
+\int_{0}^{h(x)}V(x-y+\Delta_T)dV^{*n}(y)\nonumber\\
& &+P(S_{n+1}\in x+\Delta_T, S_n>h(x), X_{n+1}>h(x))\nonumber\\
&=&I_1(x)+I_2(x)+I_3(x).\label{308}
\end{eqnarray}
{When $x\ge x_1$, by (\ref{305}), we have}
\begin{eqnarray}
I_1(x)({V}(x+\Delta_T))^{-1}&=&\int_{0}^{h(x)}
{{V^{*n}}(x-y+\Delta_T)}({V}(x+\Delta_T))^{-1}dV(y)\nonumber\\
&\leq& A_n\int_{0}^{h(x)}
{{V}(x-y+\Delta_T)}({V}(x+\Delta_T))^{-1}dV(y)\nonumber\\
&\leq& A_n\lo(1+8^{-1}\varepsilon\ro).\label{309}
\end{eqnarray}
Similarly, we have
\begin{eqnarray}
I_2(x)({V}(x+\Delta_T))^{-1}&\leq&1+8^{-1}\varepsilon.\label{310}
\end{eqnarray}
Finally, by (\ref{306}),
\begin{eqnarray}
I_3(x)({V}(x+\Delta_T))^{-1}
&\leq&\int_{h(x)-T}^{x-h(x)+T}{V^{*n}}(x-y+\Delta_T)({V}(x+\Delta_T))^{-1}dV(y)\nonumber\\
&\leq&A_n\int_{h(x)-T}^{x-h(x)+T}V(x-y+\Delta_T)({V}(x+\Delta_T))^{-1}dV(y) \nonumber\\
&\leq&A_n\lo(C^T(V)- 2+8^{-1}\varepsilon\ro).\label{311}
\end{eqnarray}
So when $x\geq x_1$, it follows from (\ref{308})-(\ref{311}) that
\begin{eqnarray}
A_{n+1}&\leq&\sup\limits_{x\geq
x_1}V^{*n}(x+\Delta_T)({V}(x+\Delta_T))^{-1}\nonumber\\
&\leq&1+8^{-1}\varepsilon+A_n\lo(C^T(V)-1+4^{-1}\varepsilon\ro).\label{tc1}
\end{eqnarray}
Taking $K=K(\epsilon)=:\frac{8}{3\varepsilon}$ and using
(\ref{tc1}), we get
\begin{eqnarray*}
A_{n+1}&\leq&2+
K\left(C^T(V)-1+3\cdot4^{-1}\varepsilon\right)\textcolor[rgb]{1.00,0.00,0.00}{^{n+1}}\nonumber\\
&\leq& K\left(C^T(V)-1+\varepsilon\right)^{n+1}.
\end{eqnarray*}This completes the proof of Lemma \ref{lem5}.
~~$\hfill\Box$\medskip

For a r.v. $\xi$ with a distribution $V$
and a positive and finite mean $E\xi$, denote
$$V_1(x)=\min\{1, V^I(x)E\xi\},~x\in(-\infty,\infty),$$
{which may be a defective distribution.

\begin{lemma}\label{l34} Let two distributions $V$ and $V_{1}$ be described as above.
If $V\in{\cal L}$, then
$V_1\in{\cal L}_{\d}$ for all $0<T<\infty$; on the contrary, if
$V_1\in{\cal L}_{\d}$ for some $0<T<\infty$, then $V\in{\cal L}$.
And both are able to derive the following asymptotic equivalence
formula: \be\label{q121}V_1(x+{\d})\sim \overline{V}(x)T.\ee
\end{lemma}
\proof Clearly, since $V\in{\cal L}$, $V_1\in{\cal L}_{\d}$ for all $0<T<\infty$ and
(\ref{q121}) holds. On the contrary, from the following fact that for any $y>0$ and
$x$ large enough,
\beq \overline{V}(x-y)T\le V_1(x-y-T+{\d})\sim
V_1(x+{\d})\le\overline{V}(x)T, \eeq
we know that $V\in{\cal L}$ and (\ref{q121}) holds.$\hfill\Box$\medskip

In the following, we give some new versions of Pitman's theorem.
When $T=\infty$, the result is due to Pitman (1980). To this end, we
first recall a known fact. Yu and Wang (2014) show that, if
a distribution $V\in{\cal L}\cap{\cal OS}$, then for any
$h\in\mathcal{H}(V)$,\beq
\limsup\int_{h(x)}^{x-h(x)}\overline{V}(x-y)(\overline{V}(x))^{-1}dV(y)=C^*(V)-2.\eeq
And by Lemma \ref{l31} (ii), if $V\in{\cal L}_{\d}\cap{\cal OS}_{\d}$
for some $0<T\le\infty$, then (\ref{37}) holds.

Similar to the proof of (\ref{37}), we can prove that, if a r.v. $\xi$ with distribution $V\in{\cal L}\cap{\cal {OS}^*}$, then for any
$h\in\mathcal{H}(V)$,
\be\label{q03}
\limsup\int_{h(x)}^{x-h(x)}\overline{V}(x-y)\overline{V}(y)(\overline{V}(x))^{-1}dy
=C^{\otimes}(V)-2E\xi.
\ee

\begin{lemma}\label{l35} Let $V$ and $U$ be two distributions,
then the following assertions hold.

(i) If $V\in{\cal L}_{\d}\cap{\cal
OS}_{\d}$ for some $0<T\le\infty$ and there exist two constants
$c_1$ and $c_2$ such that \be\label{q10} 0<c_1=\liminf\frac
{U(x+\d)}{V(x+\d)}\le \limsup\frac {U(x+\d)}{V(x+\d)}=c_2<\infty,\ee
then $U\in{\cal OS}_{\d}$ and \be\label{q01}C^T(U)-2c_1^{-1}c_2\le
c_1^{-1}c_2^2(C^T(V)-2).\ee Particularly, if $c_1=c_2=c_0$, then
$U\in{\cal L}_{\d}\cap{\cal OS}_{\d}$ and
\be\label{q12}C^T(U)-2=c_0(C^T(V)-2).\ee

(ii) If $V\in{\cal L\cap\cal OS^*}$ and there exist two constants
$c_1$ and $c_2$ such that \be\label{q130} -\infty<c_1=\liminf U(x+\d)(\overline{V}(x))^{-1}\le \limsup U(x+\d)(\overline{V}(x))^{-1}=c_2<\infty\ee
for some $0<T<\infty$, then $U\in{\cal OS}_{\d}$ and
\be\label{q133}c_1^2(c_2T)^{-1}(C_{\otimes}(V)-2E\xi)\le
C_T(U)-2\le C^T(U)-2\le c_2^2(c_1T)^{-1}(C^{\otimes}(V)-2E\xi).\ee
Particularly,
if $c_1=c_2=c_0$, then both are able to derive the following
equations:
\be\label{q130}C^T(U)-2=c_0T^{-1}(C^\otimes(V)-2E\xi)
~and~C_T(U)-2=c_0T^{-1}(C_\otimes(V)-2E\xi).\ee
Thus, $V\in{\cal L}\cap{\cal OS}^*$ if and only if $U\in{{\cal
L}_{\Delta_T}\cap\cal OS}_{\d}$ for all $0<T<\infty$.
Further, we have
\be\label{q13}C^T(V_1)-2=C^\otimes(V)-2E\xi~and~C_T(V_1)-2=c_0T^{-1}(C_\otimes(V)-2E\xi).\ee
\end{lemma}
\proof (i) In (\ref{308}), we take  $n=1$ and $V=U$, then for any
$h\in\mathcal{H}_{\d}(V)$,  \be\label{q1331}\limsup
I_i(x)(U(x+\d))^{-1}\le c{_1}^{-1}c_2,~i=1,2.\ee
Let r.v.s $\xi_1$ and
$\xi_2$ have distributions $V$ and $U$ respectively. Then by (\ref{q10}), we have
\beq &&\limsup\frac {I_3(x)}{U(x+\d)}\le
c_2\limsup \int_{h(x)-T}^{x-h(x)+T}\frac {V(x-y+\d)}{U(x+\d)}dU(y)\\
&=&c_2\limsup P(\xi_1+\xi_2\in x+\d, ~h(x)-T<\xi_2\le
x-h(x)+T)(U(x+\d))^{-1}\\
&\le&c_2\limsup P(\xi_1+\xi_2\in x+\d,~h(x)-2T<\xi_1\le x-h(x)+2T)(U(x+\d))^{-1}\\
&=&c_2\limsup\int_{h(x)-2T}^{x-h(x)+2T}U(x-y+\d)(U(x+\d))^{-1}dV(y)\\
&\le&c_1^{-1}c_2^2\limsup\int_{h(x)-2T}^{x-h(x)+2T}V(x-y+\d)(V(x+\d))^{-1}dV(y)\\
&=&c_1^{-1}c_2^2(C^T(V)-2).\eeq Thus, by (\ref{38}) of Lemma
\ref{l31}, (\ref{q1331}) and (\ref{37}), we know that $U\in{\cal
OS}_{\d}$ and (\ref{q01}) holds.

Particularly, if $c_1=c_2=c_0$, then
$$C^T(V)-2\le c_0^{-1}(C^T(U)-2).$$ Combining the above
inequalities and (\ref{q01}) yields the equality (\ref{q12}).

(ii) When $V\in{\cal L\cap\cal OS^*}$, similarly to (i), for some
$h\in\mathcal{H}_{\Delta_T}(U)$, we have
\be\limsup I_i(x)(U(x+\d))^{-1}\le c{_1}^{-1}c_2,~i=1,2.\ee
For $I_3(x)$, without loss of generality, we may assume that $l_1(x)=(x-2h_1(x))T^{-1}$
is {an} integer for $x$ large enough, where $h_1(x)=h(x)-T$. Thus, by (\ref{q10}), we have
\beq &&\limsup I_3(x)(U(x+\d))^{-1}\le
c_2\limsup \int_{h_1(x)}^{x-h_1(x)}\o V(x-y)(U(x+\d))^{-1}dU(y)\\
&=&c_2\limsup \sum_{k=1}^{l_1(x)}\int_{h_1(x)+(k-1)T}^{h_1(x)+kT}\overline{V}(x-y)(U(x+\d))^{-1} dU(y)\\
&\le&c_1^{-1}c_2^2\limsup\sum_{k=1}^{l_1(x)}\overline{V}(x-h_1(x)-kT)
\overline{V}(h_1(x)+(k-1)T)(\overline{V}(x))^{-1}\\
&=&c_1^{-1}c_2^2T^{-1}\limsup\int_{h_1(x)}^{x-h_1(x)}\overline{V}(x-y)\overline{V}(y)(\overline{V}(x))^{-1}dy\\
&=&c_1^{-1}c_2^2T^{-1}(C^\otimes(V)-2E\xi).\eeq Thus, $U\in{\cal
OS}_{\d}$.

On the other hand, we have
\be\label{q1332}\liminf I_i(x)(U(x+\d))^{-1}\ge c{_1}c_2^{-1},~i=1,2.\ee
And without loss of generality, we may assume that $l(x)=(x-2h(x))T^{-1}$ is {an}
integer for $x$ large enough, then
\beq &&\liminf I_3(x)(U(x+\d))^{-1}\ge
c_1\liminf \int_{h(x)}^{x-h(x)}\o V(x-y)(U(x+\d))^{-1}dU(y)\\
&=&c_1\liminf \sum_{k=1}^{l(x)}\int_{h(x)+(k-1)T}^{h(x)+kT}\overline{V}(x-y)(U(x+\d))^{-1} dU(y)\\
&\ge&c_1^2c_2^{-1}\liminf\sum_{k=1}^{l(x)}\overline{V}(x-h(x)-(k-1)T)\overline{V}(h_1(x)+(k-1)T)
(\overline{V}(x))^{-1}\\
&=&c_1^2c_2^{-1}T^{-1}\liminf\int_{h(x)}^{x-h(x)}\overline{V}(x-y)\overline{V}(y)(\overline{V}(x))^{-1}dy\\
&=&c_1^2c_2^{-1}T^{-1}(C_\otimes-2E\xi).\eeq
By the above four inequalities, we know that (\ref{q133}) holds.

From these results, the final result can be obtained
immediately.$\hfill\Box$\medskip

Now we discuss the relationship between a distribution $V$ and its
$\gamma$-transformation $V_{\gamma}$.

\begin{lemma}\label{l36} For some $\gamma>0$,
$V\in{\cal L}(\gamma)\cap{\cal OS}$ if and only if $V_{\gamma}\in{\cal L}_{\d}\cap{\cal OS}_{\d}$
for any $0<T<\infty$. And both of them imply the
following asymptotic equivalence formula:
\be\label{q1111}C^{T}(V_{\gamma})-2=(M_{\gamma}(V))^{-1}\gamma(C^{\otimes}(V)-2E\xi)=(M_{\gamma}(V))^{-1}C^{\ast}(V)-2.\ee
\end{lemma}
\proof According to Proposition 2.1 of Wang and Wang (2011), we know that
$V\in{\cal L}(\gamma)$ for some $\gamma>0$ if and only if $V_{\gamma}\in{\cal L}_{\d}$
for any $0<T<\infty$. And both of them are able to derive the
following asymptotic equivalence formula:
\be\label{q1112}
V_{\gamma}(x+\d)\sim(M_{\gamma}(V))^{-1}\gamma T
e^{\gamma x}\overline{V}(x).\ee
Select $h$ and $l$ as in the proof of Lemma 3.5. By (\ref{q1112}), we know that
\beq
&&\int_{h(x)}^{x-h(x)}\frac {\o V(x-y)\o V(y)}{\o V(x)}dy
\sim\frac{M_{\gamma}(V)}{\gamma T}\sum_{k=1}^{l(x)}\int_{h(x)+(k-1)T}^{h(x)+kT}\frac {V_{\gamma}(x-y+\d)V_{\gamma}(y+\d)}{V_{\gamma}(x+\d)} dy\\
&\sim&M_{\gamma}(V)(\gamma^{-1}\sum_{k=1}^{l(x)}
V_{\gamma}(x-h(x)-kT+\d)V_{\gamma}(h(x)+kT+\d)(V_{\gamma}(x+\d))^{-1}\\
&\sim&M_{\gamma}(V)\gamma^{-1}\int_{h(x)}^{x-h(x)}V_{\gamma}(x-y+\d)(V_{\gamma}(x+\d))^{-1}dV_{\gamma}(y)\\
&=&\gamma^{-1}\int_{h(x)}^{x-h(x)}\o V(x-y)(\o V(x))^{-1} dV(y).\eeq
Thus, $V\in\cal {OS}$ if and only if $V_{\gamma}\in{\cal OS}_{\d}$
for any $0<T<\infty$, and both of them are able to derive the (\ref{q1111}) holds. $\hfill\Box$
\medskip

Finally, we introduce Corollary 2.1 of Yu and Wang (2014).

\begin{lemma}\label{l37} Let $V\in{\cal{L}}(\gamma)\cap{\cal{OS}}$ for some $\gamma\geq0$ and
$U=\sum\limits_{n=0}^{\infty}p_nV^{*n}$, where $\{p_n:~n\geq0\}$ is
a sequence of nonnegative numbers satisfying
$\sum\limits_{n=0}^{\infty}p_n=1$. Suppose that there exists some
$\varepsilon_0>0$ such that
\begin{eqnarray}
\sum_{n=0}^{\infty}p_n(C^*(V)-M_\gamma(V)+\varepsilon_0)^n<\infty,\label{iv}
\end{eqnarray}
then $U\in{\cal{L}}(\gamma)\cap{\cal{OS}}$.

\end{lemma}

\subsection{Proofs of Theorem \ref{t31} and Proposition \ref{p31}}

In this section, we prove Theorem \ref{t31} and Proposition \ref{p31}, respectively.
\medskip

\noindent{\bf Proof of Theorem \ref{t31}}. We first prove (\ref{33}). By $F\in\cal L$, Corollary 3.1 of Wang
and Wang (2006) and Lemma 3.4, we know that
\bea\label{t2}G(x+\d)&\sim&
(1-p)p^{-1}\mu^{-1}F_1(x+\d)\nonumber\\&\sim&(1-p)p^{-1}\mu^{-1}T\o
F(x), \eea hence  $G\in{\cal{L}}_{\Delta_T}$. Thus by Corollary 1 of
Asmussen et al. (2003) and Theorem 3.1 of Yu et al. (2010), we know that
\be\label{t1}\liminf W(x+\d)(G(x+\d))^{-1}=p(1-p)^{-1}.\ee
So, (\ref{33}) follows from (\ref{t2}) and (\ref{t1}).

Next, we  prove (\ref{35}). We know from (\ref{t2}),
$F\in{\cal{L}}\cap{\cal{OS}^*}$, Lemmas 3.4 and 3.5 that
$F_1\in{\cal{L}}_{\Delta_T}\cap{\cal OS}_{\d}$ and
$G\in{\cal{L}}_{\Delta_T}\cap{\cal OS}_{\d}$ for all $0<T<\infty$.
By condition (\ref{34}) and Lemma 3.5, we have
\be\label{t3}p(C^T(G)-1)=p((1-p)p^{-1}\mu^{-1}(C^\otimes(F)-2EX_1^{+})+1)<1.\ee
It follows from  (\ref{31}),  (\ref{t3}), Lemmas \ref{lem5} and
\ref{l32} and the dominated convergence  theorem that
\bea\label{t4}
&&\limsup\frac{W(x+\d)}{G(x+\d)}\le(1-p)\sum_{n=1}^{\infty}p^n
\limsup\frac{G^{*n}(x+\d)}{G(x+\d)}\nonumber\\
&\le&(1-p)\sum_{n=1}^{\infty}p^n\sum_{k=0}^{n-1}(C^T(G)-1)^{n-1-k}\nonumber\\
&=&(1-p)\sum_{n=1}^{\infty}p^n((C^T(G)-1)^{n}-1)(C^T(G)-2)^{-1},
\eea
where, if $C^T(G)=2$, then we define
$((C^T(G)-1)^{n}-1)(C^T(G)-2)^{-1}=n$ by continuity. By (\ref{t3})
and (\ref{t4}), we obtain that
\beq\limsup\frac{W(x+\d)}{G(x+\d)}&\le&(1-p)(C^T(G)-2)^{-1}\lo(\frac{p(C^T(G)-1)}
{1-p(C^T(G)-1)}-\frac{p}{1-p}\ro)\nonumber\\
&=&p(1-p(C^T(G)-1))^{-1},\eeq thus (\ref{35}) follows from
(\ref{t2}), (\ref{t3}) and (\ref{q13}).

Now, we show that $W\in{\cal
L}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T}$ for any $0<T<\infty$. By
(\ref{33}), (\ref{34}), (\ref{35}) and Lemma 3.5, we immediately get
$W\in{\cal{OS}}_{\Delta_T}$. Next, we prove  $W\in{\cal
L}_{\Delta_T}$ for any $0<T<\infty$. To this end, we denote the
$-\gamma$-transform of $W$ and $G$ by $U=W_{-\gamma}$ and
$V=G_{-\gamma}$, respectively. From (\ref{31}), we know that
\be\label{3001}M_{-\gamma}(W)=(1-p)(1-p_{1})^{-1}>1,\ee thus by
(\ref{3001}), we have
\be\label{3002}M_{\gamma}(V)=(M_{-\gamma}(G))^{-1}<1\ee
and
\be\label{3003}\overline{U}(x)=(1-p_{1})\sum_{n=1}^{\infty}p_{1}^n\overline{V^{*n}}(x),\ee
where $0<p_{1}=pM_{-\gamma}(G)<1$ for some $\gamma$ large enough. By
condition (\ref{t2}), Lemma 3.5 (ii) and (\ref{34}), we have
\be\label{3004}C^{T}(G)-2=(1-p)p^{-1}\mu^{-1}(C^{\otimes}(F)-2EX_{1}^{+})<(1-p)p^{-1}.\ee
From (\ref{3004}) and Lemma 3.6, we know that
\beq p_{1}(C^{\ast}(V)-M_{\gamma}(V))<1.\eeq
Thus, by (\ref{3003}) and Lemma 3.7, $W_{-\gamma}\in\cal L(\gamma)$. According to Proposition
2.1 of Wang and Wang (2011), we know that $W\in{\cal L}_{\Delta_T}$
for any $0<T<\infty$.

Finally, if $F\in(\cal L\cap{\cal{OS^*}})\setminus \cal {S}^*$,
then by Corollary 3.2 of Wang et al. (2007), we have $W\in({\cal L}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T})\backslash{\cal{S}}_{\Delta_T}$
for any $0<T<\infty$.$\hfill\Box$
\medskip

\noindent{\bf Proof of Proposition \ref{p31}.} Let $Y$ be a random variable with distribution $G\in\cal{L}\cap\cal{OS}^*$ supported on $(-\infty,\infty)$ and finite mean $EY=-\lambda<0$. If $C^\otimes(G)<\lambda+2EY^+$, we take distribution $F=G$ and $\mu=\lambda$, then distribution $F$ satisfy the condition (\ref{34}). Otherwise, if $C^\otimes(G)\ge\lambda+2EY^+$, we set random variable $X=Y-a$ with distribution $F$ and mean $\mu=-EX$ for some $a$ large enough such that $C^\otimes(G)-2EY^+<\lambda+a$. It is easy to find that for any $h\in{\cal{H}}(F)={\cal{H}}(G)$,
\beq
C^\otimes(F)-2EX^+&=&\limsup\int_{h(x)}^{x-h(x)}\overline{F}(x-y)\overline{F}(y)dy(\overline{F}(x))^{-1}\nonumber\\
&=&\limsup\int_{h(x)}^{x-h(x)}\overline{G}(x-y)\overline{G}(y)dy(\overline{G}(x))^{-1}\nonumber\\
&=&C^\otimes(G)-2EY^+<\lambda+a=\mu,
\eeq
that is condition (\ref{34}) holds. $\hfill\Box$

\section{Some applications}
\setcounter{equation}{0}\setcounter{thm}{0}\setcounter{Corol}{0}\setcounter{lemma}{0}\setcounter{re}{0}

In this section, we give some practical applications of the results, concepts and methods of this paper to the renewal risk model and $M/G/1$ queue, respectively.

\subsection{On the ruin distribution}
In the renewal risk model, the claim sizes $Y_i,i\ge1$ are independent, identically distributed r.v.s with a common
non-degenerate distribution $F_{1}$ supported on $(0,\infty)$ and finite mean $EY_1$; the claims occur at the random instants of time $0<T_1<T_2<...$ a.s, and the inter-arrival times $Z_i,i\ge1$ are also independent, identically distributed r.v.s with a common
non-degenerate distribution $F_{2}$ supported on $[0,\infty)$ and finite mean $EZ_1$; the sequences $\{Y_i:i\ge1\}$ and $\{Z_i:i\ge1\}$ are independent of each other; the initial capital and the premium income rate are denoted by $x\ge0$ and $c>0$, respectively; finally, the net profit condition $cEZ_1>EY_1$ is required.

We denote r.v.s $X_i=Y_i-cZ_i,i\ge1$, which have a common distribution $F$ supported on $(-\infty,\infty)$ and mean $EX_1$. Then the distribution $W$ of the supremum $M$ of a random walk $\{S_n=\sum_{i=0}^n X_i:n\ge0\}$ is called the ruin distribution of the renewal risk model, and its tail distribution
$$\overline{W}=(1-p)\sum_{n=1}^{\infty}p^n\overline{G^{*n}}$$
is called the ruin probability, where $0<p=P(\tau_+<\infty)<1$ and $G=pF_+$. In particular, if $Z_1$ is exponentially distributed with intensity $(EZ_1)^{-1}$, then $p=EY_1(cEZ_1)^{-1}$ and $G=F^I_{1}$. See, for instance, Veravebeke (1977) or Embrechts et al. (1997).

Clearly, (\ref{31}) holds for any $0<T\le\infty$ in this model. Further, we assume that $F\in\mathcal{L}\cap\mathcal{OS}^*$ and condition (\ref{34}) is satisfied, then by Theorem \ref{31}, we can get the local asymptotic estimation for the ruin distribution $W$. In other words, if the local ruin probability is defined by $W(x+\Delta)$ for any $0<T<\infty$, then its asymptotic estimation is obtained.

\subsection{On the stationary distribution of the virtual waiting-time}
For the sake of simplicity, we omit the detailed description of the $M/G/1$ queue. As Kl\"{u}ppelberg (1989) pointed out, the arrival rate and the service-time distribution were denoted by $\eta$ and $F_1$ respectively, where $F_1$ has a finite mean $\mu(F_1)$. If $p=\eta\mu(F_1)<1$, then the stationary distribution $W$ of the virtual waiting-time can be written as
$$ W=(1-p)\sum_{n=0}^{\infty}p^n G^{*n},$$
where $G=F_1^I$. Thus (\ref{31}) holds for any $0<T\le\infty$.

Further, we assume that $F_1\in\mathcal{L}\cap\mathcal{OS}^*$ and $C^\otimes(F_1)<(1+p^{-1})\mu(F_1)$, then for any $0<T<\infty$, $G(x+\Delta)\sim T(\mu(F_1))^{-1}\overline{F_1}(x)$.
Therefore, by the proof of Theorem \ref{31} and Lemma \ref{l35} (ii), we can also get the local asymptotic estimation for the stationary distribution $W$ of the virtual waiting-time as follows:
\beq
Tp(1-p)^{-1}\big(\mu(F_1)\big)^{-1}&=&\liminf W(x+\Delta)\big(\overline{F_1}(x)\big)^{-1}\nonumber\\
&\le&\limsup W(x+\Delta)\big(\overline{F_1}(x)\big)^{-1}\nonumber\\
&\le& Tp\Big(1-p\big(\mu(F_1)\big)^{-1}\big(C^\otimes(F_1)-\mu(F_1)\big)\Big)^{-1}(\mu(F_1))^{-1}.
\eeq
Particularly, if $F_1\in\mathcal{S}^*$, that is $C^\otimes(F_1)=2\mu(F_1)$, then we have
$$W(x+\Delta)\big(\overline{F_1}(x)\big)^{-1}\sim Tp(1-p)^{-1}\big(\mu(F_1)\big)^{-1}.$$

In addition to the above two examples, in a number of areas of applied probability, such as branching processes, infinitely divisible distribution and so on, the distribution of the research objects can be written in the form of (\ref{31}), or the form of other compound distributions, so the results of this paper can also be applied to these areas. Here, we omit the details.

\section{Proofs of Examples}
\setcounter{equation}{0}\setcounter{thm}{0}\setcounter{Corol}{0}\setcounter{lemma}{0}

\subsection{Proof of $G_m\in{\cal{S}}\cap{\cal{OS^*}}\setminus{\cal{S^*}}$ in Example \ref{e201}.}

It is not hard to see that for all $m\geq 1$ and $x\geq x_1$, one has
{\begin{eqnarray} x^{-2\alpha+m^{-1}}\leq\overline{F}(x)\leq
2^{\alpha}x^{-\alpha}.\label{203}
\end{eqnarray}

Since $\alpha\in (m^{-1}, 1+m^{-1})$, by $\alpha>1$ and (\ref{203}), we know that
the distribution $G_m$ has a finite mean $m(G_m)=\mu$. Denote \beq
f(x)=\sum\limits_{n=1}^{\infty}(x_n^{-\alpha-1}-x_n^{-2\alpha-1+m^{-1}})\textbf{1}(x_n<x<
2x_n).\eeq

We first prove that  $G_m\notin{\cal{S^*}}$. To this end, for all
$x\geq0$, denote
\begin{eqnarray}\label{402}
H(x)&=&(\overline{G_m}(x))^{-1}\int_0^{x}\overline{G_m}(x-y)\overline{G_m}(y)dy
\end{eqnarray}

By (\ref{201}), one has
\begin{eqnarray}
&&H(2x_n)= 2(\overline{F}^m(2x_n))^{-1}\int_{x_n}^{2x_n}
\overline{F}^m(2x_n-y)\overline{F}^m(y)dy\nonumber\\
&=&2\int_{0}^{x_n}\overline{F}^m(y)(1+(x_n^{\alpha-m^{-1}-1}-x_n^{-1})y)^mdy\nonumber\\
&=&2\int_{0}^{x_n}\overline{F}^m(y)dy+2\int_{0}^{x_n}\overline{F}^m(y)
\Big(\big(1+(x_n^{\alpha-m^{-1}-1}-x_n^{-1})y\big)^m-1\Big)dy.\label{206}
\end{eqnarray}
By (\ref{203}) and (\ref{201}),
\begin{eqnarray}
\lim\limits_{n\to\infty}x_n^{\alpha
m-m-1}\int_{0}^{x_n}\overline{F}^m(y)y^mdy
&=&\lim\limits_{n\to\infty}x_n^{\alpha m-m-1}\int_{2x_n-1}^{x_n}\overline{F}^m(y)y^mdy\nonumber\\
&=&(m+1)^{-1}. \label{207}
\end{eqnarray}
And for all $t=1,\cdots,m-1$,
\begin{eqnarray}
\lim\limits_{n\to\infty}x_n^{(\alpha-m^{-1}-1)t}\int_{0}^{x_n}\overline{F}^m(y)y^tdy=0.\label{208}
\end{eqnarray}
By (\ref{206})-(\ref{208}),
\begin{eqnarray}
\lim_{n\to\infty}H(2x_n)=2\mu+2(m+1)^{-1},\label{209}
\end{eqnarray}
thus $G_m\notin{\cal{S^*}}$.

Next, we prove $G_m\in{\cal{OS^*}}$.
We estimate $H(x)$ in the cases $x_n\leq x<\frac{3}{2}x_n$,
$\frac{3}{2}x_n\leq x< 2x_n$ and $2x_n\leq x< x_{n+1},~n\ge 1$,
respectively. When $x\in[x_n,3\cdot2^{-1}x_n)$, by (\ref{203}) and
(\ref{201}),
\begin{eqnarray}
H(x)&\leq&2^{m\alpha+1}\lo(\overline{F}^m\lo({3\cdot2^{-1}x_n}\ro)\ro)^{-1}
\int_{\frac{x}{2}}^{x}\overline{F}^m(x-y)y^{-m\alpha}dy\nonumber\\
&\leq&2^{2m\alpha+1+m}\int_{0}^{3\cdot4^{-1}x_n}\overline{F}^m(y)dy\le2^{2m\alpha+1+m}\mu.\label{211}
\end{eqnarray}
When $x\in[3\cdot2^{-1}x_n,2x_n)$, by (\ref{201}), (\ref{206}) and (\ref{209}),

\begin{eqnarray}
H(x)&=& 2(\overline{F}^m(x))^{-1}\Big(\int_{\frac{x}{2}}^{x_n}+\int_{x_n}^{x}\Big)
\overline{F}^m(x-y)\overline{F}^m(y)dy\nonumber\\
&=&2(\overline{F}^m(x))^{-1}\Big(x_n^{-m\alpha}\int_{x-x_n}^{2^{-1}x}\overline
F^m(x-y)\overline F^m(y)dy+\int_{0}^{x-x_n}\overline
F^m(x-y)\overline F^m(y)dy\Big)\nonumber\\
&\leq&2(\overline{F}^m(2x_n))^{-1}x_n^{-m\alpha}\int_{2^{-1}x_n}^{x_n}\overline{F}^m(y)dy\nonumber\\
& &+2\int_0^{x-x_n}\overline{F}^m(y)\Big(1+(x_n(1-x_n^{-\alpha+m^{-1}})^{-1}
-(x-x_n))^{-1}y\Big)^mdy\nonumber\\
&\leq&1+2\int_0^{x_n}\overline{F}^m(y)
\Big(1+(x_n^{\alpha-m^{-1}-1}-x_n^{-1})y\Big)^mdy\nonumber\\
&=&1+H(2x_n)\rightarrow1+2\mu+2(m+1)^{-1},~~n\rightarrow\infty.\label{212}
\end{eqnarray}
When $x\in[2x_n,x_{n+1})$, by (\ref{201}) and (\ref{206}),
\begin{eqnarray}
H(x)&\leq& 2(\overline{F}^m(x))^{-1}\Big(\int_{x_n}^{2x_n}
+\int_{2x_n}^{x}\Big)\overline{F}^m(x-y)\overline{F}^m(y)dy\nonumber\\
&\leq&H(2x_n)+2\int_0^{x-2x_n}\overline{F}^m(y)dy
\rightarrow4\mu+2(m+1)^{-1},~~~n\rightarrow\infty.\label{213}
\end{eqnarray}
It follows from (\ref{211})-(\ref{213}) that $G_m\in{\cal{OS^*}}$.

Finally, we prove $G_m\in{\cal{S}}$. By (\ref{203}), we have
\begin{eqnarray}
\overline{F}^{2m}(2^{-1}x)(\overline{F}^m(x))^{-1}
&\leq&2^{4m\alpha}x^{-1}\to0.\label{214}
\end{eqnarray}
By (\ref{214}) and
\begin{eqnarray*}
\overline{G_m^{*2}}(x)=2\overline{G_m}(x)-\overline{G_m}^2(2^{-1}x)
+2\int_{2^{-1}x}^{x}\overline{G_m}(x-y)dG_m(y),
\end{eqnarray*}
we know that in order to prove  $G_m\in{\cal{S}}$, it suffices to
prove
\begin{eqnarray}
T(x)&=&2(\overline{F}^m(x))^{-1}\int_{2^{-1}x}^{x}\overline{F}^m(x-y)d\lo(1-\overline{F}^m(x)\ro)\nonumber\\
&=&2m(\overline{F}^m(x))^{-1}\int_{2^{-1}x}^{x}\overline{F}^m(x-y)\overline{F}^{m-1}(y)f(y)dy\to0.\label{215}
\end{eqnarray}
Clearly, $T(x_n)=0,n\geq1$. By (\ref{201}) and (\ref{208}), we have
\begin{eqnarray}
&&T(2x_n)=2m\overline{F}^m(2x_n)\int_{0}^{x_n}\overline{F}^m(y)
\overline{F}^{m-1}(2x_n-y)f(2x_n-y)dy\nonumber\\
&=&2m(x_n^{\alpha-m^{-1}-1}-x_n^{-1})\int_{0}^{x_n}
\overline{F}^m(y)(1+(x_n^{\alpha-m^{-1}-1}-x_n^{-1})y)^{m-1}dy\nonumber\\
&\leq&2mx_n^{\alpha-m^{-1}-1}\int_{0}^{x_n}\overline{F}^m(y)
(1+x_n^{\alpha-m^{-1}-1}y)^{m-1}dy\nonumber\\
&=&2mx_n^{\alpha-m^{-1}-1}\Big(\int_{0}^{x_n}\overline{F}^m(y)dy+
\int_{0}^{x_n}\overline{F}^m(y)((1+x_n^{\alpha-m^{-1}-1}y)^{m-1}-1)dy\Big)\nonumber\\
&\to&0, ~~n\to\infty.\label{217}
\end{eqnarray}
In the following, we prove (\ref{215}) in the cases $x_n\leq x< 2x_n$
and $2x_n\leq x< x_{n+1},~n\ge 1$, respectively. When $x\in[x_n,2x_n)$,
by (\ref{201}) and (\ref{217}),
\begin{eqnarray}
&&T(x)=  2m(\overline{F}^m(x))^{-1}\int_{x_n}^{x}\overline{F}^m(x-y)
\overline{F}^{m-1}(y)f(y)dy\nonumber\\
&=&2m(\overline{F}^m(x))^{-1}\int_{0}^{x-x_n}\overline{F}^m(y)
\overline{F}^{m-1}(x-y)f(x-y)dy\nonumber\\
&\leq& 2m(x_n^{\alpha-m^{-1}-1}-x_n^{-1})\int_{0}^{x-x_n}
\overline{F}^m(y)\Big(1+(x_n(1-x_n^{-\alpha+m^{-1}}
)^{-1}-(x-x_n))^{-1}y\Big)^{m-1}dy\nonumber\\
&\leq& T(2x_n)\to0, ~~n\to\infty.\label{218}
\end{eqnarray}
When $x\in[2x_n,x_{n+1})$, by (\ref{201}) and (\ref{217}),
\begin{eqnarray}
T(x)&\leq&2m(\overline{F}^m(2x_n))^{-1}\int_{x_n}^{2x_n}
\overline{F}(2x_n-y)\overline{F}^{m-1}(y)f(y)dy\nonumber\\
&=& T(2x_n)\to0, ~~n\to\infty.\label{219}
\end{eqnarray}
According to (\ref{218}) and (\ref{219}), (\ref{215}) holds, thus
$G_m\in{\cal{S}}$.

In summary, we have
$G_m\in({\cal{S}}\cap{\cal{OS^*}})\setminus{\cal{S^*}}$.\hfill$\Box$
\medskip

\subsection{ Proof of
$G_m\in({\cal{L}}\cap{\cal{OS}^*})\setminus{\cal{S}}$ in Example
\ref{e202}.}

Still let $f$ be the density of $F$, when $x\geq x_1,$
it is easily seen that
\begin{eqnarray}
x^{-2^{-1}(\alpha+1)}\leq\overline{F}(x)\leq
2^{\alpha}x^{-2^{-1}\alpha}
\mbox{~and~}f(x)\leq2^{\alpha+1}x^{-2^{-1}\alpha-1}\label{220}
\end{eqnarray}

Moreover, one can easily find that the distribution $G_m$ has a
finite mean for $m\geq 1$, and we still denote it by $\mu.$

Firstly, by (\ref{220}) we have,
\begin{eqnarray*}
(\overline F(x))^{-1}f(x)\leq2^{\alpha+1}x^{-2^{-1}}\to0, ~~ ,
\end{eqnarray*}
thus $F\in{\cal{L}}$, so $G_m\in{\cal{L}}$. Next, we prove that
$G_m\in{\cal{OS}^*}.$ Let $H(x), x\geq0$ be the same as in
(\ref{402}), it is easily seen that
\begin{eqnarray*}
H(4x_n^2)&=&2(\overline{F}^m(4x_n^2))^{-1}\int_{0}^{2x_n^2}\overline{F}^m(y)\overline{F}^m(4x_n^2-y)dy\nonumber\\
&=&2\int_{0}^{2x_n^2}\overline F^m(y)(1+(1-x_n^{-1})(4x_n^2-y)^{2^{-1}}+2x_n)^{-1}y)^mdy\nonumber\\
&\leq&2\int_{0}^{2x_n^2}\overline F^m(y)(1+(2x_n)^{-1}y)^mdy\nonumber\\
&\leq&2x_1^2(1+(2x_n)^{-1}x_1^2)^m+2^{m\alpha+1}
\int_{x_1^2}^{\infty}y^{m-\frac{m\alpha}{2}}(y^{-1}+(2x_n)^{-1})^mdy\nonumber\\
&\leq&2x_1^2(1+x_1^2)^m+2^{m\alpha+1}(2^{-1}m\alpha-m-1)^{-1}x_1^{2m-m\alpha}(1+x_1^2)^m<\infty.
\end{eqnarray*}
Just as in  Example \ref{e201}, we deal with $H(x)$ in three cases
$x_n^2\leq x<2x_n^2$, $2x_n^2\leq x< 4x_n^2$ and
$4x_n^2\leq x< x_{n+1}^2,~n\ge 1$, respectively. When $x\in[x_n^2,2x_n^2)$,
just as (\ref{211}), by (\ref{202}) and variable substitution, we
have
\begin{eqnarray}
H(x)&=&2(\overline F^{m}(x))^{-1}\int_{\frac{x}{2}}^{x}\overline F^{m}(x-y)\overline F^{m}(y)dy\nonumber\\
&\leq&2(\overline F^{m}(2x_n^2))^{-1}\int_{\frac{x}{2}}^{x}\overline F^{m}(x-y)x_n^{-m\alpha}dy\nonumber\\
&\leq&2(2-2^{2^{-1}})^{-m}\int_{0}^{x_n^2}\overline F^{m}(y)dy<\infty.\label{222}
\end{eqnarray}
When $x\in[2x_n^2,4x_n^2)$, just as (\ref{212}), by (\ref{202}), we
have

\begin{eqnarray}
&&H(x)=2(\overline F^{m}(x))^{-1}\int_{0}^{2^{-1}x}\overline F^{m}(y)\overline F^{m}(x-y)dy\nonumber\\
&=&2(\overline F^{m}(x))^{-1}\int_{0}^{2^{-1}x}\overline F^{m}(y)
\Big(\overline F(x)+(x_n^{-\alpha-1}-x_n^{-\alpha-2})(x^{2^{-1}}+(x-y)^{2^{-1}})^{-1}y\Big)^mdy\nonumber\\
&=&2\int_{0}^{2^{-1}x}\overline F^{m}(y)\Big(1+(x_n^{-\alpha-1}-x_n^{-\alpha-2})\big(\overline F(x)(\sqrt{x}+\sqrt{x-y})\big)^{-1}y\Big)^mdy\nonumber\\
&\leq&2\int_{0}^{2x_n^2}\overline F^{m}(y)(1+(2^{2^{-1}}x_n)^{-1}y)^mdy\nonumber\\
&\leq&2x_1^2(1+(2^{2^{-1}}x_n)^{-1}x_1^2)^m+2^{m\alpha+1}\int_{x_1^2}^{\infty}y^{m-\frac{m\alpha}{2}}(y^{-1}
+(2^{2^{-1}}x_n)^{-1})^mdy\nonumber\\
&\leq&2x_1^2(1+x_1^2)^m+2^{m\alpha+1}(2^{-1}m\alpha-m-1)^{-1}x_1^{2m-m\alpha}(1+x_1^2)^m<\infty.\label{223}
\end{eqnarray}
When $x\in[4x_n^2,x_{n+1}^2)$, just as (\ref{213}), by (\ref{202}),
we have
\begin{eqnarray}
H(x)&\leq&2(\overline F^{m}(x))^{-1}\lo(\int_{2x_n^2}^{4x_n^2}+\int_{4x_n^2}^{x}\ro)\overline F^{m}(x-y)\overline F^{m}(y)dy\nonumber\\
&\leq&H(4x_n^2)+2\int_{0}^{x-4x_n^2}\overline F^{m}(y)dy<\infty.\label{224}
\end{eqnarray}
According to (\ref{222}) and (\ref{223}), (\ref{224}) holds, that is
$G_m\in{\cal{OS^*}}$.

Finally, we prove  $G_m\notin{\cal{S}}$. Since
\begin{eqnarray*}
\overline{G_m}^2(2x_n^2)(\overline{G_m}(4x_n^2))^{-1}&=&x_n^{m-m\alpha}
\big((2-2^{2^{-1}})+(2^{2^{-1}}-1)x_n^{-1}\big)^{2m}\to0,~~n\to\infty.
\end{eqnarray*}
Therefore, in order to prove  $G_m\notin{\cal{S}}$, we only need to prove
\begin{eqnarray}
\liminf_{ }T(x)=\liminf_{x\rightarrow
\infty}(\overline{G_m}(x))^{-1}\int_{2^{-1}x}^{x}\overline{G_m}(x-y)G_m(dy)
>0.\label{225}
\end{eqnarray}
In fact,
\begin{eqnarray*}
&&T(4x_n^2)=m(\overline{F}^m(4x_n^2))^{-1}\int_{2x_n^2}^{4x_n^2}\overline{F}^m(4x_n^2-y)\overline{F}^{m-1}(y)f(y)dy\nonumber\\
&=&2^{-1}m(1-x_n^{-1})\int_{0}^{2x_n^2}\overline{F}^m(y)
\Big(1+(1-x_n^{-1})(\sqrt{4x_n^2-y}+2x_n)^{-1}y\Big)^{m-1}y^{-2^{-1}}dy\nonumber\\
&\geq&2^{-1}m(1-x_n^{-1})\int_{x_1^2}^{2x_n^2}\overline{F}^m(y)y^{-\frac{1}{2}}dy\nonumber\\
&\geq&m(m\alpha+m-1)^{-1}(1-x_n^{-1})\big(x_1^{1-m\alpha-m}-(2x_n^2)^{-2^{-1}(m\alpha+m-1)}\big)\nonumber\\
&\to& m(m\alpha+m-1)^{-1}x_1^{1-m\alpha-m}>0,~~n\to\infty,
\end{eqnarray*}
that is, (\ref{225}) holds, that is $G_m \notin{\cal{S}}$.

In summary, we have
$G_m\in({\cal{L}}\cap{\cal{OS^*}})\setminus{\cal{S}}$.\hfill$\Box$

\subsection{ Proof of
$G_m^I\in({\cal{L}}_{\Delta_T}\cap{\cal{OS}}_{\Delta_T})\setminus{\cal{S}}_{\Delta_T}$
in Example \ref{e203}.}

This conclusion follows directly from
$G_m\in({\cal{S}}\cap{\cal{OS^*}})\setminus{\cal{S^*}}$, Lemmas 3.4,
3.5 and Lemma 4.2 of Wang et al. (2007).\hfill$\Box$

\subsection{ Proof of $G_m^I\in{\cal{OS}}_{\Delta_T}\setminus{\cal{S}}_{\Delta_T}$ for all
$0<T<\infty$ in Example \ref{e204}.}

By Proposition 2.1 of  Chen et al. (2013), we have $F_{\gamma}\in
{\cal{L}}_{\Delta_T}\backslash{\cal{S}}_{\Delta_T}$ for all $T>0$.
We now prove $F_{\gamma}\in {\cal{OS}}_{\Delta_T}$.  Since
$F_i\in{\cal{S}}(\gamma)\subset{\cal{OS}},~i=1,2$, so
$F=F_1*F_2\in{\cal{OS}}$ by Proposition 6.1 of Yu and Wang (2013).
By $F\in{\cal{OS}}$ and (2.1) of  Chen et al. (2013),  we have
\begin{eqnarray*}
\limsup F_{\gamma}^{*2}(x+\Delta_T)(F_{\gamma}(x+\Delta_T))^{-1}=
(M_{\gamma}(F))^{-1}\limsup\overline{F^{*2}}(x)(\overline{F}(x))^{-1}<\infty,
\end{eqnarray*}
thus  $F_{\gamma}\in {\cal{OS}}_{\Delta_T}$.\hfill$\Box$

\subsection{Proofs of $G_m\notin\cal{OS}^*$,
$G_m^I\in(\cal{L}\cap{\cal{OS}})\setminus{\cal{S}}$ and
$G_m^I\in{\cal{L}}_{\Delta_T}\setminus{\cal{OS}}_{\Delta_T}$ for all $0<T<\infty$ in Example
\ref{e205}.}

To show that $G_m\notin\cal{OS}^*$, we denote the
density of $F$ by $f$. Consider the following quantity
\begin{eqnarray*}
I(x)=\big(\overline{G_m}(2x_n)\big)^{-1}\int_{x_n}^{2x_n-2x_{n-1}}\overline{G_m}(2x_n-y)\overline{G_m}(y)dy.\nonumber\\
\end{eqnarray*}
Since  $x_n\leq y\leq {2x_n-2x_{n-1}}\leq 2x_n$, we have
$2x_{n-1}\leq {2x_n-y}\leq x_n,~n\ge 1$, so by (\ref{227}),
\begin{eqnarray*}
I(x)&=&2^{2m\alpha}x_n^{m\alpha}\int_{x_n}^{2x_n-2x_{n-1}} (x_n^{-\alpha}-f(x_n)(y-x_n))^m dy\nonumber\\
&=&2^{2m\alpha}(m+1)^{-1}x_n^{m\alpha}(x_n-2x_{n-1})\sum_{i=0}^{m}x_n^{-\alpha i}(x_n^{-\alpha}-f(x_n)(x_n-2x_{n-1}))^{m-i}\\
&\geq&2^{2m\alpha}(m+1)^{-1}(x_n-2x_{n-1})\rightarrow\infty,~~n\rightarrow\infty,
\end{eqnarray*}
thus  $G_m\notin{\cal{OS}}^*$.

Since $G_m\in{\cal{L}}\setminus{\cal{OS}^*}$, we immediately get
$G_m^I\in{\cal{L}}_{\Delta_T}\setminus{\cal{OS}}_{\Delta_T}$ by
Lemmas 3.4 and 3.5.

Now we show that $G_m^I\notin{\cal{S}}$. By (\ref{227}), it is
obvious that

{\begin{eqnarray} x^{-2\alpha}\leq\overline{F}(x)\leq
x^{-\alpha}.\label{228}
\end{eqnarray}}
By (\ref{228}), we have
\begin{eqnarray*}
(\overline{G_m^I}(x_n))^2(\overline{G_m^I}(2x_n))^{-1}&\geq&
\mu^{-1}\Big(\int_{x_n}^{2x_n}\overline{F}^m(y)dy\Big)^2
\Big(\int_{2x_n}^{\infty}\overline{F}^m(y)dy\Big)^{-1}\nonumber\\
&\geq&\mu^{-1}(m+1)^{-2}2^{2m\alpha-2}(1-m^{-1}\alpha^{-1})>0,
\end{eqnarray*}
thus by (2.4) of  Murphree (1989), we complete the proof.

To prove  $G_m^I\in{\cal{OS}}$, we consider

\begin{eqnarray}
&&\int_{\frac{x}{2}}^{x}\overline{G_m^I}(x-y)dG^I_m(y)=
\mu^{-1}  \int_{\frac{x}{2}}^{x}\overline{F}^m(y)\int_{x-y}^{\infty}\overline{F}^m(z)dzdy\nonumber\\
&=&\mu^{-1}\bigg(\int_{\frac{x}{2}}^{x}\overline{F}^m(y)\int_{x-y}^{\frac{x}{2}}\overline{F}^m(z)dzdy
+\Big(\int_{\frac{x}{2}}^{x}\overline{F}^m(y)dy\Big)^2
+\int_{\frac{x}{2}}^{x}\overline{F}^m(y)\int_{x}^{\infty}\overline{F}^m(z)dzdy\bigg)\nonumber\\
&=&\mu^{-1}(I_1(x)+I_2(x)+I_3(x)).\label{229}
\end{eqnarray}

We first estimate $I_1(x)$ in the cases $x_n\leq x<4x_n$ and $4x_n\leq
x< x_{n+1},~n\ge 1$, respectively.  When $x\in [x_n,4x_n)$, by
(\ref{227}) and (\ref{228}), we have
\begin{eqnarray}
&&\Big(\int_{x}^{\infty}\overline{F}^m(y)dy\Big)^{-1}I_1(x)
\leq\Big(\int_{4x_n}^{x_{n+1}}\overline{F}^m(y)dy\Big)^{-1}
\int_{2^{-1}x}^{x}x_n^{-m\alpha}\int_{x-y}^{2^{-1}x}\overline{F}^m(z)dzdy\nonumber\\
&\leq&2^{2m\alpha-2}(x_n^{2-m\alpha}-x_n^{1-m\alpha})^{-1}\int_0^{2^{-1}x}
\int_{y}^{2^{-1}x}\overline{F}^m(z)dzdy\nonumber\\
&\leq&2^{2m\alpha-1}(x_n^{2-m\alpha})^{-1}\Big(\int_{x_1}^{2x_n}\int_{y}^{2x_n}z^{-m\alpha}dzdy
+\int_{0}^{x_1}\int_{x_1}^{2x_n}z^{-m\alpha}dzdy
+\int_{0}^{x_1}\int_{y}^{x_1}dzdy\Big)\nonumber\\
&\leq&2^{m\alpha+1}(m\alpha-1)^{-1}(2-m\alpha)^{-1}+((m\alpha-1)^{-1}x_1^{2-m\alpha}+x_1^2)<\infty.\label{230}
\end{eqnarray}
Similarly,  when $x\in [4x_n,x_{n+1})$, we have
\begin{eqnarray}
&&\Big(\int_{x}^{\infty}\overline{F}^m(y)dy\Big)^{-1}I_1(x)
\leq\Big(\int_{x_{n+1}}^{2x_{n+1}}\overline{F}^m(y)dy\Big)^{-1}
\int_{2^{-1}x}^{x}(2x_n)^{-2m\alpha}\int_{x-y}^{2^{-1}x}\overline{F}^m(z)dzdy\nonumber\\
&\leq&
\frac{m+1}{2x_n}\Big(\int_{x_1}^{\frac{x_{n+1}}{2}}\int_{y}^{\frac{x_{n+1}}{2}}z^{-m\alpha}dzdy+
\int_{0}^{x_1}\int_{x_1}^{\frac{x_{n+1}}{2}}z^{-m\alpha}dzdu+
\int_{0}^{x_1}\int_{y}^{x_1}dzdy\Big)\nonumber\\
&\leq&(m+1)((m\alpha-1)^{-1}(2-m\alpha)^{-1}2^{-m\alpha}
+((m\alpha-1)^{-1}x_1^{2-m\alpha}+x_1^2)<\infty.\label{231}
\end{eqnarray}
Next, we estimate $I_2(x)$. When $x\in[x_n,x_{n+1})$, by (\ref{228}), we have
\begin{eqnarray}
&&\Big(\int_{x}^{\infty}\overline{F}^m(y)dy\Big)^{-1}\Big(\int_{2^{-1}x}^{x}\overline{F}^m(y)dy\Big)^2
\leq\Big(\int_{x_{n+1}}^{2x_{n+1}}\overline{F}^m(y)dy\Big)^{-1}
\Big(\int_{2^{-1}x_n}^{x_{n+1}}\overline{F}^m(y)dy\Big)^2\nonumber\\
&\leq&(m+1)\Big(2^{-4+2m\alpha}9+3x_n^{1-m\alpha}+(2x_n)^{2-2m\alpha}\Big)<\infty.\label{232}
\end{eqnarray}
Finally, for $I_3(x)$, it is obvious that
\begin{eqnarray}
\Big(\int_{x}^{\infty}\overline{F}^m(y)dy\Big)^{-1}I_3(x)
&\leq&\int_{2^{-1}x}^{x}\overline{F}^m(y)dy\to0.\label{233}
\end{eqnarray}
By (\ref{229})-(\ref{233}), we get $G_m^I\in{\cal{OS}}$.\hfill$\Box$

\vspace{0.2cm}

{\bf Acknowledgement.}
The authors are grateful to the two referees for his/her
careful reading and valuable comments and suggestions, which greatly improve the original version of this paper.

config: arxiv


TITLE: What sort of manifold is PU(H)?
QUESTION [4 upvotes]: The space underlying the projective unitary group of a separable, infinite-dimensional Hilbert space has a number of topologies, so for the purposes of this question, pick you favourite and answer for that one.
I've read that $PU(H)$ is a Fréchet manifold, but that was without saying which topology. There are two ways I can see to think about this. First is that if we know what sort of manifold $U(H)$ is, then we know what $PU(H)$ is, as the former looks locally like a chart of $PU(H)$ times $U(1)$. So we can consider the closed (I think!) subspace $U(H) \subset End(H)$.
Alternately we can consider $PU(H)$ as sitting inside the Hilbert-Schmidt operators on $H$ as the projective unitaries act freely on the latter. (EDIT: this is not right, as pointed out by Andrew. I was thinking of the inclusion $PU(H) \hookrightarrow U(HS(H))$, which doesn't really tell us much in hindsight.)
I'm not familiar enough with the analysis to turn the above observations into results, and I may be interested in other topologies.
So the question is, is there the structure of a Banach or even Hilbert manifold on $PU(H)$?

REPLY [2 votes]: Edit: Theo's comments are spot-on - that'll teach me to try to post a quick answer without thinking too much
There are two sensible topologies on $U(H)$, each leads to a topology on $P U(H)$ by quotienting:

The norm topology.  In this topology, $U(H)$ is viewed as a sub-Lie group of the Banach Lie group $Gl(H)$.  Here, $Gl(H)$ is open in $B(H)$ and the exponential map is a diffeomorphism onto a neighbourhood of the identity and this takes skew-Hermitian operators onto $U(H)$, thus defining charts.  As the quotient $U(H) \to P U(H)$ admits slices, this makes $P U(H)$ a Banach manifold.
The weak topology.  In this topology, we start with $Gl(H)$ as a subspace of $B(H)$ where this has the weak topology in which $A_\lambda \to A$ if $A_\lambda v \to A v$ for each $v \in H$.  However, this isn't good enough for $Gl(H)$ as taking the inverse isn't continuous.  So we put on it the topology so that both the inclusion $Gl(H) \to B(H)$ and the inclusion after the inversion map $A \to A^{-1}$ are continuous.  However, then $Gl(H)$ is not an open subspace of $B(H)$ so it isn't obviously a manifold.  Indeed, I don't think that with this topology then it is a manifold at all (I don't think it is an ANR).  As Theo says, the induced topology on $U(H)$ doesn't need the inversion fix, but even so it still is not a manifold.

The map to Hilbert-Schmidt operators is not useful here because the action is not free, it is only faithful.  It acts by conjugation whereupon the stabiliser group of a particular element are all the unitary elements that commute with it and this can be quite large.


text:
\begin{document}
\title[
 Inverse source problem for advection-diffusion]
{A novel method to solve inverse source problem for advection-diffusion equation from final data}

\author[Z. Li
]
{Zhiyuan Li, \quad Gongsheng Li, \quad Xiangzheng Jia}

\address{Zhiyuan Li, \quad Gongsheng Li, \quad Xianzheng Jia \newline
School of Mathematics and Statistics,
Shandong University of Technology.
266 Xincunxi Road, Zibo, Shandong 255049, China}
\email{zyli@sdut.edu.cn,\ ligs@sdut.edu.cn,\ jxz\_1@sdut.edu.cn}

\thanks{Submitted \today.}
\subjclass[2010]{35K20, 35R30, 35B53}
\keywords{advection-diffusion; inverse problem; final observation; uniqueness; \hfill\break\indent  Liouville's theorem.
}

\begin{abstract}
In this article, for an advection-diffusion equation we study an inverse problem for restoration of source temperature from the information of final temperature profile. The uniqueness of this inverse problem is established by taking an integral transform and using Liouville's theorem (complex analysis).
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{lemma}{Lemma}[section]
\newtheorem{theorem}{Theorem}

\allowdisplaybreaks

\section{Introduction and main results}
Throughout this paper, assuming $T$ is a given positive number, and $\Omega\subset\mathbb R^d$ is a bounded domain with a sufficiently smooth boundary $\partial\Omega$, which is defined e.g., by some $C^2$ functional relations, we study the following initial-boundary value problem
\begin{equation} \label{eq}
\left\{
\begin{alignedat}{2}
&\partial_t u - L u = \rho(t)f(x), &\quad&  (x,t) \in Q, \\
&u(x,0)=0,&\quad& x\in \Omega, \\
&u(x,t)=0, &\quad&  (x,t)\in \partial\Omega\times(0,T],
\end{alignedat}
\right.
\end{equation}
where $Q:=\Omega\times(0,T]$, $\rho$ is known as a $C^1[0,T]$-function, and $L$ is a second-order linear differential operator from $H^2(\Omega)\cap H_0^1(\Omega)$ to $L^2(\Omega)$ which can be written in the following form
$$
Lu:= \sum_{i,j=1}^d \partial_i(a_{ij}(x)\partial_j u) + \sum_{j=1}^d b_j(x)\partial_j u + c(x)u,\quad u\in H^2(\Omega)\cap H_0^1(\Omega),
$$
where $c,b_j\in L^\infty(\Omega)$, $j=1,\cdots,d$, and $a_{ij}=a_{ji}\in C^1(\overline\Omega)$, $1\le i,j\le d$ are given functions. Moreover there exists a positive constant $a_0 > 0$ such that
$$
\sum_{i,j=1}^d a_{ij}(x) \xi_i\xi_j \ge a_0 |\xi|^2,\quad \forall x\in\overline\Omega,\ \forall\xi:=(\xi_1,\cdots,\xi_d)\in\mathbb R^d.
$$

The differential equation in \eqref{eq} is called an advection-diffusion equation and is used as model equation for the description of density fluctuations in a material undergoing diffusion. It is also used to describe processes exhibiting diffusive-like behavior, for instance the \lq diffusion' of alleles in a population in population genetics (see, e.g., \cite{AW75}). We also refer to several fields of applications such as hydrology \cite{B72}, chemistry \cite{BN91}, and also in spatial ecology to modelize the dynamic of a population \cite{M02}. There are also many works from theoretical aspects and here we do not intend to give a complete list of references, and the readers can consult Chapter 7 in \cite{Evans} and \cite{F08} for example.

In \eqref{eq}, the term $\rho(t)f(x)$ models a source term, and it is crucial to know or determine the coefficient $f$. For example, \cite{CC04} pointed out that in population dynamic, the extinction of the population, its persistence and its evolution is conditioned by this term $f$. However, usually this term cannot be directly measured due to the mixing of the effects of several factors, which requires
one to use inverse problems to identify these quantities by involving additional information that can be observed or measured practically. In other words, our main concern is:

{\bf Inverse source problem.} Let us assume that $\rho$ is known. Determine the unknown source magnitude $f$ in $\Omega$ from the final state observation
$$
u(x,T),\quad \mbox{$x\in\Omega$.}
$$

Similar kinds of inverse problems have been studied by many authors. For $b_j\equiv0$, $j=1,\cdots,d$, \cite{HS13} and \cite{R80} established a uniqueness of the inverse problem in determining the spacewise-dependent source by the use of semi-group theory. For a more general source term $\rho$ which is $x$- and $t$-dependent, as is seen from Theorem 3.1 in \cite{I91}, the assumption $\rho(x,t)>0$ in $\overline Q$ is not sufficient to guarantee the uniqueness of the inverse problem. Actually, in this case, we need to assume $\partial_t \rho(x,t)>0$ in $\overline Q$. However, under a further assumption $\partial_t\rho>0$, \cite{I91} employed the strong maximum principles for the elliptic and parabolic equations to derive the uniqueness. There are some other tricks to overcome the ill-posedness of this inverse problem, we refer to \cite{CY96} and \cite{CY99} where the diffusion equation was parametrized by a diffusion coefficient and a stability inequality was established except for a countable set of parameters provided that $\rho(\cdot,T)>0$ in $\overline\Omega$, and refer to \cite{BC16} in which the Carleman estimate was used to recover the source from the measurement of the temperature at fixed time provided that the source is known in an arbitrary subdomain.

In this paper, for the advection-diffusion equation in \eqref{eq}, we show that in the case where $\rho$ is only $t$-dependent, the assumption $\rho>0$ in $[0,T]$ is enough to determine the unknown source term from the final overdetermination. We have

\begin{theorem}
\label{thm-unique}
Assume $\rho\in C^1[0,T]$ is positive for any $t\in[0,T]$. Let $f\in L^2(\Omega)$ and suppose $u$ solves the initial-boundary value problem \eqref{eq}. Then $u(\cdot,T)\equiv0$ in $\Omega$ implies $f\equiv0$ in $\Omega$.
\end{theorem}

Here we should mention that in the case where $b_j=0$ and $c\ge0$, we can derive the uniqueness and stability results in the framework of the Fourier expansion. Indeed, by using the orthonormality of the Dirichlet eigensystem of elliptic operator $L$, the unknown source term $f$ can be represented by the final measurement. It will be difficult to follow this argument to derive the uniqueness in the general case. On the other hand, it is also very difficult to follow the semigroup arguments used in \cite{HS13} and \cite{R80} because of the nonsymmetric term.

The key idea of our proof is to transfer the advection-diffusion equation to an elliptic one by using an integral transform with respect to $t$. Several technical lemmas for this elliptic equation are needed, so we collect them in Section \ref{sec-unique}. Preparing all necessities, say, Lemmas \ref{lem-rho} and \ref{lem-analytic}, by Liouville's theorem (complex analysis) we will finish the proof of Theorem \ref{thm-unique}. Finally, concluding remarks are given in Section \ref{sec-rem}.

\section{Proof of Theorem \ref{thm-unique}}\label{sec-unique}

In this section, we will give a proof of our main result. For this, we multiply both sides of the differential equation in \eqref{eq} by $e^{-(s+M)t}$ with $s\in\mathbb C$ and $M>0$, and integrate from $0$ to $T$ to derive
\begin{equation}
\label{trans-u}
\int_0^T \partial_t u(t) e^{-(s+M)t} dt - \int_0^T L u(t) e^{-(s+M)t} dt = f(x)\int_0^T \rho(t) e^{-(s+M)t} dt.
\end{equation}
From integration by parts, and noting that $u(0)=u(T)=0$, we further calculate the first term on the left hand side of the above identity as follows
$$
\int_0^T \partial_t u(t) e^{-(s+M)t} dt
=(s+M)\int_0^T  u(t) e^{-(s+M)t} dt.
$$
For short, we put
\begin{equation}
\label{eq-trans-u}
\widehat u(s):=\int_0^T  u(t) e^{-(s+M)t} dt,\quad \widehat\rho(s):=\int_0^T\rho(t) e^{-(s+M)t} dt.
\end{equation}
Therefore \eqref{trans-u} can be rephrased by
\begin{equation}
\label{eq-lap-u}
- L \widehat u(s) + (s+M)\widehat u(s) = f(x)\widehat\rho(s),\quad s\in\mathbb C.
\end{equation}
Next we are to give an estimate for $\widehat u(s)$ for $s\in\mathbb C$ on the basis of the above equation. We first introduce a useful lemma which gives a lower bound of the function $|\widehat \rho(s)|$ for some $s\in\mathbb C$.
\begin{lemma}
\label{lem-rho}
Let $N>1$ be a sufficiently large and fixed constant. Assuming the function $\rho\in C^1[0,T]$ is positive in $[0,T]$. Then there exists a positive constant $C=C(\inf_{[0,T]} \rho=:\rho_0,T,N)$ such that the following inequality
$$
|\widehat \rho(s)| = \left|\int_0^T \rho(t) e^{-st} dt \right| \ge C\int_0^T e^{-t\Re s} dt
$$
is valid for any $s\in\{s\in\mathbb C;\ \Re s\le 0,\ \Im s\ge0,\ N\Im s+\Re s\le0\}$.
\end{lemma}
\begin{proof}
Firstly, by integration by parts we represent $\widehat \rho(s)$ as follows
$$
\int_0^T \rho(t) e^{-st} dt
= \frac{\rho(0) - \rho(T) e^{-sT} }{s}
+ s^{-1} \int_0^T \rho'(t) e^{-st} dt
=:I_1(s) + I_2(s).
$$
For $I_2(s)$, by a direct calculation, we find
$$
|I_2(s)|
\le |s|^{-1} \int_0^T |\rho'(t)| e^{-t\Re s} dt
\le \frac{\|\rho\|_{C^1[0,T]}}{|s|}  \frac{1 - e^{-T\Re s}}{\Re s}.
$$
Moreover, the choice of $s$, say, $\Re s\le0$, $\Im s\ge0$ and $N\Im s+\Re s\le0$ imply that $N\Im s\le |\Re s|$, hence that
\begin{equation}
\label{esti-s-Res}
|s| = \sqrt{(\Re s)^2 + (\Im s)^2} \le |\Re s| + |\Im s| \le (1+1/N) |\Re s|
\le 2 |\Re s|,\quad N>1,
\end{equation}
which can be further used to derive that
$$
|I_2(s)|
\le \frac{2\|\rho\|_{C^1[0,T]}( e^{T|\Re s|} - 1)}{|s|^2}
\le \frac{4\|\rho\|_{C^1[0,T]} e^{T|\Re s|}}{|s|^2}.
$$
Let us turn to evaluating $I_1(s)$. For this, a direct calculation derives
$$
sI_1(s) = \rho(0) - \rho(T) e^{-T\Re s} \cos(T\Im s)  + i \rho(T) e^{-T\Re s} \sin(T\Im s),
$$
hence that
$$
|sI_1(s)|^2 = \rho(T)^2 e^{-2T\Re s} -2\rho(0)\rho(T) e^{-T\Re s} \cos(T\Im s) + \rho(0)^2,
$$
finally that
$$
|I_1(s)|^2 = \frac{\left(\rho(T) e^{-T\Re s} - \rho(0)\right)^2}{|s|^2} + 2\rho(0) \rho(T) e^{-T\Re s} \frac{1- \cos(T\Im s)}{|s|^2}.
$$
Moreover, by the use of the inequality
$$
\frac{|1-\cos t|}{t^2} \le C,\quad t\in\mathbb R,
$$
we see that
$$
\frac{|1- \cos(T\Im s)|}{|s|^2}
=\frac{|1- \cos(T\Im s)|}{|T\Im s|^2} \frac{|T\Im s|^2}{|s|^2} \le CT^2,\quad s\in\mathbb C.
$$
Combining all the above estimates, we find
$$
|I_1(s)|^2 \ge \frac{\left(\rho(T) e^{-T\Re s} - \rho(0)\right)^2}{|s|^2} - 2CT^2\rho(0) \rho(T) e^{-T\Re s}.
$$
Now for sufficiently large $|s|$, then $|\Re s|$ is also sufficiently large in view of \eqref{esti-s-Res}. Consequently there exists a constant $C=C(T,\rho)$ such the estimate
$$
|I_1(s)| \ge \frac{Ce^{-T\Re s}}{|s|}.
$$
privided that $|s|>K$ with $K>0$ is a sufficiently large constant. Finally, we have
$$
\left|\int_0^T \rho(t) e^{-st} dt\right|
\ge |I_1(s)| - |I_2(s)|
\ge \frac{e^{T|\Re s|}}{|s|} \left(C - \frac{4\|\rho\|_{C^1[0,T]}  }{|s|} \right)
\ge \frac{Ce^{T|\Re s|}}{2|s|}
$$
for $|s|>K$ satisfying $\Re s\le0$, $\Im s\ge0$ and $N\Im s + \Re s\le0$, where $N\ge1$.

Next, we evaluate $\widehat \rho(s)$ in the case when $s$ lies in $\{s\in\mathbb C; |s|\le K,\ \Re s\le0,\ \Im s\ge0, N\Im s + \Re s\le0\}$. By taking sufficiently large $N$, we are led to
$$
|\Im s| \le \frac{|\Re s|}{N} \le \frac{K}{N},
$$
which implies that
$$
|\sin (t\Im s)| \le |t\Im s| \le \frac{TK}{N} \le \frac14,
$$
and
$$
\cos (t\Im s) \ge \frac{1}2,
$$

We first rewrite $\widehat \rho(s)$ as follows
$$
\widehat\rho(s) = \int_0^T \rho(t) e^{-t\Re s} \cos (t\Im s) dt + i \int_0^T \rho(t) e^{-t\Re s} \sin (t\Im s) dt,
$$
hence that
$$
|\widehat\rho(s)| \ge \frac12 \int_0^T \rho(t) e^{t|\Re s|}  dt - \frac14 \int_0^T \rho(t) e^{t|\Re s|} dt
\ge\frac{\rho_0}4 \int_0^T e^{t|\Re s|} dt
= \frac{\rho_0(e^{T|\Re s|}- 1)}{4|\Re s|},
$$
where in the last inequality we used the assumption $\rm A1$.

Finally, we obtain
$$
|\widehat \rho(s)| \ge C\frac{e^{T|\Re s|} - 1}{|\Re s|}>C_1>0,\quad s\in\{s\in\mathbb C;\Re s\le0, \Im s\ge0, N\Im s+ \Re s\le0\},
$$
where the constant $C_1>0$ is only dependent on $T,\rho,N$.
\end{proof}

\begin{lemma}
\label{lem-analytic}
Under the assumptions in Theorem \ref{thm-unique}, the function $\widehat u: \mathbb C\to H^2(\Omega)\cap H_0^1(\Omega)$ which is defined by \eqref{eq-trans-u} can be estimated as follows
\begin{equation}
\label{esti-u}
\|\widehat u(s)\|_{L^2(\Omega)}
\le C|\widehat \rho(s)| \|f\|_{L^2(\Omega)},\quad s\in \mathbb C.
\end{equation}
\end{lemma}
\begin{proof}
We denote $\overline{\widehat u}(s)$ as the conjugate of the complex valued function $\widehat u(s)$. Then multiplying $\overline{\widehat u}(s)$ on both sides of the equation \eqref{eq-lap-u}, and taking integral on $\Omega$ lead to
$$
\int_\Omega (-L\widehat u(s)) \overline{\widehat u}(s) + (s+M)\widehat u(s) \overline{\widehat u}(s) dx
= \int_\Omega \widehat \rho(s) \overline{\widehat u}(s) dx,\quad
s\in\mathbb C.
$$
Here we will omit the variable $x$ in $\widehat u(x,s)$ when no confusion can arise. Integration by parts yields
\begin{equation}
\label{eq-var}
B_s[\widehat u(s),\widehat u(s)] = \int_\Omega \widehat \rho(s) \overline{\widehat u}(s) dx,
\end{equation}
where
$$
B_s[\widehat u(s),\widehat u(s)]
=\int_\Omega \sum_{i,j=1}^d a_{ij} (\partial_j \widehat u(s)) \partial_i \overline{\widehat u}(s)
+ \sum_{j=1}^d b_j (\partial_j \widehat u(s)) \overline{\widehat u}(s) + (s+c+M) |\widehat u(s)|^2 dx.
$$
Then the symmetry of $\{a_{ij}\}$ implies that $\sum_{i,j=1}^d a_{ij} \partial_j \widehat u(s) \partial_i \overline{\widehat u}(s) \in\mathbb R$ for any $s\in\mathbb C$, hence the real part of $B_s[\widehat u,\widehat u] $ can be represented as follows
\begin{equation}
\label{eq-re}
\Re\left(B_s[\widehat u,\widehat u] \right)
=\int_\Omega \sum_{i,j=1}^d a_{ij} (\partial_j \widehat u) \partial_i \overline{\widehat u}
+ \Re\left[\sum_{j=1}^d b_j (\partial_j \widehat u) \overline{\widehat u}\right] + (\Re s+c+M) |\widehat u|^2 dx,
\end{equation}
while the imaginary part is
\begin{equation}
\label{eq-im}
\Im\left(B_s[\widehat u,\widehat u] \right)
=\int_\Omega\Im\left[\sum_{j=1}^d b_j (\partial_j \widehat u) \overline{\widehat u}\right] + (\Im s) |\widehat u|^2 dx.
\end{equation}
Now in the case of $\Re s\ge0$, from \eqref{eq-re}, we find
\begin{equation}
\label{esti-re}
\int_\Omega \sum_{i,j=1}^d a_{ij} (\partial_j \widehat u) \partial_i \overline{\widehat u}
+ (\Re s+c+M) |\widehat u|^2 dx
\le |B_s[\widehat u,\widehat u] | + \left|\sum_{j=1}^d b_j (\partial_j \widehat u) \overline{\widehat u} \right|,
\end{equation}
where we used the inequality $\Re s\le |\Re s|\le |s|$, if $s\in\mathbb C$. Since $c\in L^\infty(\Omega)$, we can choose $M>0$ large enough so that $\Re s+ c + M\ge M/2$ for $\Re s\ge0$. Moreover, the ellipticity of $\{a_{ij}\}$ and the use of Poincar\'e's inequality imply
$$
\int_\Omega \sum_{i,j=1}^d a_{ij} (\partial_j \widehat u) \partial_i \overline{\widehat u}
\ge a_0 \| \nabla \widehat u\|_{L^2(\Omega)}^2
\ge C\|\widehat u\|_{H^1(\Omega)}^2.
$$
Here and henceforth, $C>0$ is regarded as a generic constant which only depends on $d,\Omega, a_{ij},b_j,c,\rho$. Consequently, in view of \eqref{eq-var}, from the H\"{o}lder inequality we derive
\begin{align*}
&C\|\widehat u(s)\|_{H^1(\Omega)}^2
+ \frac{M}2 \|\widehat u(s)\|_{L^2(\Omega)}^2
\\
\le& |\widehat \rho(s)| \|f\|_{L^2(\Omega)} \|\widehat u(s)\|_{L^2(\Omega)}
+ C\|\nabla \widehat u(s)\|_{L^2(\Omega)} \|u(s)\|_{L^2(\Omega)},\quad \Re s\ge0,
\end{align*}
and hence
\begin{align*}
&C\|\widehat u(s)\|_{H^1(\Omega)}^2
+ \frac{M}2 \|\widehat u(s)\|_{L^2(\Omega)}^2
\\
\le& |\widehat \rho(s)| \|f\|_{L^2(\Omega)}\|\widehat u(s)\|_{L^2(\Omega)}
+ \varepsilon C^2\|\nabla \widehat u(s)\|_{L^2(\Omega)}^2 + \frac{1}{4\varepsilon} \|u(s)\|_{L^2(\Omega)}^2,\quad \Re s\ge0.
\end{align*}
Now we choose $\varepsilon>0$ and $M>0$ such that
$$
\varepsilon C^2 \le \frac{C}2,\quad \frac{M}2\ge \frac{1}{4\varepsilon},
$$
which implies that
\begin{align*}
\|\widehat u(s)\|_{H^1(\Omega)}
\le \frac{2}{C}|\widehat \rho(s)| \|f\|_{L^2(\Omega)},\quad \Re s\ge0.
\end{align*}
Next, if $s\in \{\Re s\le0; N\Im s + \Re s\ge0\}$, where $N>0$, again by the fact \eqref{eq-var} and the H\"older inequality, we evaluate the imaginary part of $B_s[\widehat u(s),\widehat u(s)]$ as follows
\begin{equation}
\label{esti-im}
(\Im s)  \int_\Omega |\widehat u(s)|^2 dx
\le  |\widehat \rho(s)| \|f\|_{L^2(\Omega)} \|\widehat u(s)\|_{L^2(\Omega)}
+ C\|\nabla\widehat u(s)\|_{L^2(\Omega)} \|\widehat u(s)\|_{L^2(\Omega)},
\end{equation}
Similarly, \eqref{eq-re} can be further treated by
\begin{align*}
&C\|\widehat u(s)\|_{H^1(\Omega)}^2
+ (\Re s+c+M) \|\widehat u(s)\|_{L^2(\Omega)}^2
\\
\le& |\widehat \rho(s)| \|f\|_{L^2(\Omega)} \|\widehat u(s)\|_{L^2(\Omega)}
+ C\|\nabla\widehat u(s)\|_{L^2(\Omega)} \|\widehat u(s)\|_{L^2(\Omega)},
\end{align*}
Collecting all the above estimates, it follows that
\begin{align*}
&C\|\widehat u(s)\|_{H^1(\Omega)}^2
+ (N\Im s + \Re s+c+M) \|\widehat u(s)\|_{L^2(\Omega)}^2
\\
\le& (N+1)|\widehat \rho(s)| \|f\|_{L^2(\Omega)} \|\widehat u(s)\|_{L^2(\Omega)}
+ C(N+1)\|\nabla\widehat u(s)\|_{L^2(\Omega)} \|\widehat u(s)\|_{L^2(\Omega)}
\\
\le& (N+1)|\widehat \rho(s)| \|f\|_{L^2(\Omega)}\|\widehat u(s)\|_{L^2(\Omega)}
+ \varepsilon C^2(N+1)^2\|\nabla\widehat u(s)\|_{L^2(\Omega)}^2 + \frac{1}{4\varepsilon}\|\widehat u(s)\|_{L^2(\Omega)}
,
\end{align*}
Now from the choice of $s$, say, $s\in \{\Re s\le0; N\Im s + \Re s\ge0\}$, we see that $N\Im s + \Re s+c+M\ge\frac{M}2$ provided that $M>0$ is sufficiently large. Moreover, by choosing $\varepsilon>0$ such that $\varepsilon C^2(N+1)^2 \le \frac{C}2$ and $\frac{M}2\ge \frac1{4\varepsilon}$, we arrive at the following inequality
\begin{align*}
\frac{C}2\|\widehat u(s)\|_{H^1(\Omega)}^2
\le (N+1)|\widehat \rho(s)| \|f\|_{L^2(\Omega)}\|\widehat u(s)\|_{L^2(\Omega)},
\end{align*}
that is
\begin{align*}
\|\widehat u(s)\|_{H^1(\Omega)}
\le \frac{2(N+1)}{C}|\widehat \rho(s)| \|f\|_{L^2(\Omega)},\quad \Re s\le0,\ N\Im s+ \Re s\ge0, \ N>0.
\end{align*}

Let us turn to considering the case when $s\in\{\Re s\le0; \Im s\ge0; N\Im s+\Re s\le0\}$ with a sufficiently large $N>0$. From the previous estimate for the function $\rho$ in Lemma \ref{lem-rho}, we have
$$
\|\widehat u(s)\|_{H^1(\Omega)}
\le \int_0^T \|u(t)\|_{H^1(\Omega)} e^{-t\Re s}dt
\le C|\widehat \rho(s)| \|f\|_{L^2(\Omega)}.
$$
Collecting all the above estimates, we finally obtain
$$
\|\widehat u(s)\|_{H^1(\Omega)}  \le C|\widehat \rho(s)| \|f\|_{L^2(\Omega)},\quad s\in\mathbb C,
$$
which completes the proof of the lemma.
\end{proof}

Now we are ready to give a proof of our main theorem.

\begin{proof}[\bf Proof of Theorem \ref{thm-unique}]
Since $\widehat u(s)$ and $\widehat \rho(s)$ are analytic in $\mathbb C$, and hence $\frac{\widehat u(s)}{\widehat \rho(s)}$ is a holomorphic function in $\mathbb C$ removing the zero points of $\widehat\rho(s)$. However, Lemma \ref{lem-analytic} and Riemann's theorem (see, e.g., \cite{R87}) guarantee that all the singularities of the function $\frac{\widehat u(s)}{\widehat \rho(s)}$ are removable, and then this function can be regarded as an analytic function in $\mathbb C$ and is bounded on $\mathbb C$. Thus the Liouville theorem (see, e.g., \cite{R87}) implies that the analytic function $\frac{\widehat u(s)} {\widehat \rho(s)}$ must be independent of $s$, that is, there exists a function $\phi(x)$ such that
$$
\phi(x) = \frac{\widehat u(x;s)} {\widehat \rho(s)},\quad s\in\mathbb C.
$$
We claim that $\phi$ must vanish in $\Omega$. Indeed, by using integration by parts, and noting that $u(0)=u(T)=0$, we find that
$$
s\widehat u(s) = s\int_0^T u(t) e^{-st}dt
=\int_0^T u'(t) e^{-st}dt,\quad s>0,
$$
hence that
$$
|s\widehat u(s)| \le \|u\|_{C^1[0,1]} \int_0^T e^{-st}dt
\le \frac{\|u\|_{C^1[0,1]}(1-e^{-sT})}{s}\to 0,\ \mbox{ as $s\to+\infty$.}
$$
Similarly
$$
s\widehat \rho(s) = \rho(0) - \rho(T)e^{-sT} + \int_0^T u'(t) e^{-st}dt
\to \rho(0),\ \mbox{ as $s\to+\infty$.}
$$
Since $\phi$ is independent of $s$, these two estimates for $s\widehat u(s)$ and $s\widehat\rho(s)$ imply that $\phi$ is vanished in $\Omega$, finally that
$$
\widehat u(x;s)\equiv0,\quad x\in\Omega,\ s\in\mathbb R
$$
in view of the fact
$$
\widehat \rho(s) = \int_0^T \rho(t) e^{-st} dt > 0,\quad s\in\mathbb R,
$$
hence that
$$
\int_0^\infty \widetilde u(t) e^{-st} dt = 0,\quad s\in\mathbb R,
$$
where $\widetilde u(t):=u(t)$ if $t\in(0,T)$ and $\widetilde u$ vanishes outside of $(0,T)$, which implies $u(x,t)\equiv0$ in $\Omega\times(0,T)$ due to the uniqueness of the Laplace transform, and hence $f\equiv0$ in $\Omega$. This completes the proof of the main result.

\end{proof}

\section{Concluding remarks}
\label{sec-rem}

In this paper, the initial-boundary value problem \eqref{eq} where the source term takes the form of separated variables was considered. In Theorem \ref{thm-unique} we proved that the spatial component in the source term can be uniquely determined by the final observation data. The key idea is to employ an integral transform with respect to the temporal variable to transfer the advection-diffusion equation to the corresponding elliptic equation with a parameter. On the basis of several estimates for this elliptic equation (see, Lemmas \ref{lem-rho} and \ref{lem-analytic}), we finished the proof by using Liouville's theorem (complex analysis). It should be mentioned here that the proof of the above uniqueness result heavily relies on the setting that all the coefficients in the operator $L$ is $t$-independent. It would be interesting to investigate what happens about this inverse problem in the framework of $(x,t)$-dependent coefficients.

\section*{Acknowledgement}
The author thanks Grant-in-Aid for Research Activity Start-up 16H06712, Japan Society of the Promotion of Science (JSPS).