text:
\begin{document}

\author{Salim BOUZEBDA$^{*}$ \&  Amor KEZIOU$^{**}$}
\address{$^{*}$LSTA-Universit\'{e} Paris 6, 175, rue du Chevaleret,
Bo\^{i}te 158, 75013 Paris, bouzebda@ ccr.jussieu.fr}

\address{$^{**}$Laboratoire de Math\'ematiques (UMR 6056) CNRS,
Universit\'e de Reims Champagne-Ardenne and LSTA-Universit\'e
Paris 6. UFR Sciences, Moulin de la Housse, B.P. 1039,
                   51687 Reims,
                   France.
amor.keziou@univ-reims.fr
  }

\date{February  2007.}

\title{New estimates and tests of
independence in some copula models}

\maketitle

\begin{abstract}{\it We introduce new estimates and tests of
independence in copula models with  unknown margins using
$\phi$-divergences and the duality technique. The asymptotic laws
of the estimates and the test statistics are established both when
the parameter is an interior point or not.}\\

\noindent Key words: Dependence function ;  Multivariate rank
statistics; Semiparametric inference; Boundary value parameter.
\end{abstract}
\section{Introduction and motivations}
\noindent Parametric models for copulas with unknown margins have
been intensively investigated during the last decades.
\emph{Copulas} have become popular in applied statistics, because
of the fact that they constitute a flexible and robust way to
model dependence between the margins of random vectors. In this
framework, semiparametric inference methods, based on
\emph{pseudo-likelihood}, have been applied to copulas by a number
of authors (see, e.g., \cite{Shih_Louis1995},
\cite{Wang_Ding2000}, \cite{tsukahara2005} and the references
therein). Throughout the available literature, investigations on
the asymptotic properties of parametric estimators, as well as the
relevant test statistics, have privileged the case where the
parameter is an interior point of the admissible domain. However,
for most parametric copula models of interest, the boundaries of
the admissible parameter spaces include some important parameter
values, typically among which, that corresponding to the
independence of margins. We find in \cite{Joe1997} many examples
of parametric copulas, for which marginal independence is verified
for some specific values of the parameter $\theta$, on the
boundary $\partial \Theta$ of the admissible parameter set $\Theta
\subseteq \mathds{R}^d$, $d\geq 1$.  This paper concentrates on
this specific problem. We aim, namely, to investigate parametric
inference procedures, in the case where the parameter belongs to
the boundary of the admissible domain. In particular, it will
become clear, that the usual limit laws both for parametric copula
estimators and test statistics become invalid under these limiting
cases, and, in particular, under marginal independence. Motivated
by this observation, we will introduce a new semiparametric
inference procedure based on \emph{$\phi$-divergences} and the
\emph{duality} technique. We will show that the proposed
estimators remain asymptotically normal, even under the marginal
independence assumption for appropriate choice of the divergence.
This will allow us to introduce test statistics of independence,
whose study will be made, both under the null and the alternative
hypotheses. Let $F(x_{1},x_{2}):=P(X_1\leq x_1,X_2\leq x_2)$ be a
$2$-dimensional distribution function, and $F_{i}(x_i) := P(X_{i}
\leq x_i),~i=1,2$ the marginal distribution of $F$. It is well
known since the work of  \cite{Sklar1959} that there exists a
distribution  function $C$ on $[0,1]^2$ with uniform marginals
such that
\begin{gather*} C(u_{1},u_{2}) :=
P\left\{F_{1}(X_{1}) \leq u_{1},F_{2}(X_{2}) \leq u_{2}\right\}.
\end{gather*} See also \cite{deheuvels1979b},
\cite{Deheuvels1981b,deheuvels1981}, \cite{Moore1975} and
\cite{Schweizer1991}. Formally, copulas can be defined as follows.
\begin{definition}
An $p$-dimensional copula is a function $C:[0,1]^p\rightarrow
[0,1]$ with the following properties
\begin{enumerate}
\item $C$ is grounded, i.e., for every
$\mathbf{u}=(u_1,\ldots,u_d)$, $C(\mathbf{u})=0$ if at least one
coordinate $u_i=0,~i=1,\ldots,p$; \item $C$, is $p$-increasing,
i.e., for every $\mathbf{u}\in[0,1]^p$ and $\mathbf{v}\in[0,1]^p$
such that $\mathbf{u}\leq \mathbf{v}$, the $C$-volume
$V_C[\mathbf{u}, \mathbf{v}]$ of the box $[\mathbf{u},
\mathbf{v}]$ is non negative (see \cite{Nelsen1999}); \item
$C(1,\ldots,1,u_i,1,\ldots,1)=u_i$ for all $u_i\in[0,1]^p$,
$u_i=0,~i=1,\ldots,p$.
\end{enumerate}
\end{definition}
\noindent Many useful multivariate models for dependence between
$X_1$ and $X_2$ turn out to be generated by \emph{parametric}
families of copulas of the form
$\left\{C_\theta;~\theta\in\Theta\right\}$, typically indexed by a
vector valued parameter $\theta\in\Theta\subseteq\mathds{R}^d$
(see, e.g., \cite{Kimeldorf1075_1}, \cite{Kimeldorf1075_2},
\cite{Nelsen1999},
 and \cite{Joe1993} among others). In the sequel, we assume that
 $C_\theta(\cdot,\cdot)$ admits a density $c_\theta(\cdot,\cdot)$
 with respect to the Lebesgue measure $\lambda$ on $\mathds{R}^2$.
The nonparametric approach to copula estimation has been initiated
by \cite{deheuvels1979b}, who introduced and investigated the
\emph{empirical copula process}. In addition,
\cite{Deheuvels1981b,deheuvels1981} described the limiting
behavior of this empirical process (see, also
\cite{fermanianradulovicdragan2004} and the references therein).
In this paper, we consider the estimation and test problems for
semiparametric copula models with unknown general margins. Let
$(X_{1k},X_{2k})$ $k=1,\ldots,n$ be a bivariate sample with
distribution function
$F_{\theta_T}(\cdot,\cdot)=C_{\theta_T}(F_1(\cdot),F_2(\cdot))$
where $\theta_T\in\Theta$ is used to denote the true unknown value
of the parameter. In order to estimate $\theta_T$,  some
semiparametric estimation procedures, based on the maximization,
on the parameter space $\Theta$, of properly chosen
\emph{pseudo-likelihood} criterion, have been proposed by
\cite{Oakes1994}, \cite{Shih_Louis1995}, \cite{sellieng1996},
\cite{Wang_Ding2000} and \cite{tsukahara2005} among others. In
each of these papers, some asymptotic normality properties are
established for $\sqrt{n}\big(\tilde{\theta}-\theta_T\big)$, where
$\tilde{\theta}=\tilde{\theta}_n$ denotes a properly chosen
estimator of $\theta_T$. This is achieved, provided that
$\theta_T$ lies in the \emph{interior}, denoted by
$\mathring{\Theta}$, of the parameter space $\Theta\subseteq
\mathds{R}^d$. On the other hand, the case where $\theta_T\in
\partial \Theta:=\overline{\Theta}-\mathring{\Theta}$ is a
\emph{boundary value} of $\Theta$, has not been studied in a
systematical way until present. Moreover, it turns out that, for
the above-mentioned estimators, the asymptotic normality of
$\sqrt{n}\big(\tilde{\theta}-\theta_T\big),$ may fail to hold for
$\theta_T\in
\partial \Theta$. In the framework of full parametric models with i.i.d.
data, \cite{selflieng1987} summarized all the earlier work and
provided the exact limiting distributions which are complicated by
the number of unknown parameters, how many of them are on the
boundary and the correlation between the components of the
estimator proposed. Our approach is novel in this setting and
 it
will become clear later on from our results, that the asymptotic
normality of the estimate based on $\phi$-divergences holds, even
under the independence assumption, when, either, $\theta_0$ is an
interior, or a boundary point of $\Theta$, independently of the
dimension of the parameter space. The proposed test statistics of
independence using $\phi$-divergences are also studied, under the
null hypothesis $\mathcal{H}_0$ of independence, as well as under
the alternative hypothesis. The asymptotic distributions of the
test statistics under the alternative hypothesis are used to
derive an approximation to the power functions. An application of
the forthcoming results will allow us to evaluate the sample size
necessary to guarantee a pre-assigned power level, with respect to
a specified alternative.  To establish our results, we use similar
arguments as those developed by  \cite{tsukahara2005} in
connection with the instrumental statements on rank statistics
established by \cite{Ruymgaart_Shorack_Zwet1972} and
\cite{Ruschendorf1976} among others, combined with a new technique
(based on the law of iterated logarithm given in Lemma
\ref{lemma1} below) to show both existence and consistency  of our
estimates and test statistics. All mathematical developments  are
relegated to the appendix.
\section{A new inference procedure}
\noindent Recall that the $\phi$-divergences between a bounded
signed measure $Q$, and a probability $P$ on $\mathcal{D}$, when
$Q\ll P$ is absolutely continuous with respect to $P$, is defined
by
$$D_\phi(Q,P):=\int_{\mathcal{D}} \phi\left(\frac{dQ}{dP}\right)~dP,$$
where $\phi$ is a proper closed convex function from
$]-\infty,\infty[$ to $[0,\infty[$ with $\phi(1)=0$ and such that
the domain ${\rm dom}\phi:=\{x \in \mathds{R}: \phi(x)<\infty\}$
is an interval with end points $a_{\phi}<1<b_{\phi}$. The
Kullback-Leibler, modified Kullback-Leibler, $\chi^2$, modified
$\chi^2$, Hellinger and $L^1$ divergences are examples of
$\phi$-divergences; they are obtained respectively for
$\phi(x)=x\log x-x+1$, $\phi(x)=-\log x+x-1$,
$\phi(x)=\frac{1}{2}(x-1)^2$,
$\phi(x)=\frac{1}{2}\frac{(x-1)^2}{x}$, $\phi(x)=2(\sqrt{x}-1)^2$
and $\phi(x)=|x-1|.$ We refer to \cite{Liese-Vajda1987}  for a
systematic theory of divergences. In the sequel,  for all
$\theta$, we denote by $D_\phi(\theta,\theta_T)$ the
$\phi$-divergences between $C_{\theta}$ and $C_{\theta_T}$, i.e.,
\begin{equation}
D_\phi(\theta,\theta_T):=\int_I\phi\left(\frac{dC_{\theta}}{dC_{\theta_T}}\right)~dC_{\theta_T}(u_1,u_2)=
\int_I\phi\left(\frac{c_{\theta}}{c_{\theta_T}}\right)~dC_{\theta_T}(u_1,u_2).
\end{equation}
Denote $C_n$ the empirical copula associated to the data, i.e.,
\begin{equation}\label{ec}
C_n(u_1,u_2):=\frac{1}{n}\sum_{k=1}^{n}\mathds{1}_{\left\{F_{1n}(X_{1k})\leq
u_1\right\}} \mathds{1}_{\left\{F_{2n}(X_{2k})\leq
u_2\right\}},~(u_{1},u_{2})\in I,
\end{equation}
and $F_{jn}(t):= \frac{1}{n}\sum_{k=1}^n
\mathds{1}_{]-\infty,t]}(X_{jk}),~j=1,2$. In order to estimate the
divergences $D_\phi(\theta,\theta_T)$ for a given
$\theta\in\Theta$ in particular for $\theta=\theta_0$, and the
parameter $\theta_T$, we will make use of  the dual representation
of $\phi$-divergences obtained by \cite{BK2005111} Theorem 4.4 and
\cite{Keziou2003} Theorem 2.3. By this, we readily obtain that
$D_\phi(\theta_0,\theta_T)$ can be rewritten into
\begin{equation}\label{eqn 2 dual}
D_\phi(\theta_0,\theta_T):=\sup_{f\in\mathcal{F}}\left\{\int_I
f~dC_{\theta_0}-\int_I \phi^*(f)~dC_{\theta_T}\right\},
\end{equation}
where $\phi^*$ is used to denote the convex conjugate of $\phi$,
namely, the function defined by
$$\phi^* ~:~t\in\mathds{R}\mapsto \phi^*(t):=
\sup_{x\in\mathds{R}}\left\{tx-\phi(x)\right\},$$ and
$\mathcal{F}$ is an arbitrary class of measurable functions
fulfilling  the following two conditions:~ $\forall
f\in\mathcal{F},~\int |f|~dC_{\theta_0}$ is finite and
$\phi'(dC_{\theta_0}/dC_{\theta_T})\in\mathcal{F}$.

\noindent Furthermore, the sup in the above display is unique and
is achieved at $f=\phi'(dC_{\theta_0}/dC_{\theta_T}).$

\noindent Note that the plug-in estimate
$$\int_I\phi\left(\frac{dC_{\theta}}{dC_{n}}\right)~dC_{n}(u_1,u_2)$$
of $D_\phi(\theta,\theta_T)$ is not well defined since $C_\theta$
is not absolutely continuous with respect to $C_n$; the use of the
dual representation, as we will show, avoids this problem. By the
above statement, taking the class of functions
$$\mathcal{F}=\left\{u\in I\mapsto \phi'\left(1/c_\theta\right);~\theta\in\Theta\right\},$$
we obtain the formula
\begin{equation}\label{formula 1}
D_\phi(\theta_0,\theta_T)=\sup_{\theta\in\Theta}\left\{\int_I\phi'\left(\frac{1}{c_\theta}\right)du_1du_2-
\int_I\left[\frac{1}{c_\theta}\phi'\left(\frac{1}{c_\theta}\right)-\phi\left(\frac{1}{c_\theta}\right)\right]dC_{\theta_T}(u_1,u_2)\right\},
\end{equation} whenever
$\int_I\left|\phi'\left(1/c_\theta\right)\right|~du_1du_2$ is
finite for all $\theta\in\Theta$. Furthermore, the sup is unique
and reached at $\theta=\theta_T$. Hence, the divergence
$D_\phi(\theta_0,\theta_T)$ and the parameter $\theta_T$ can be
estimated respectively by
\begin{equation}\label{estimateur 1 div }
\sup_{\theta\in\Theta}\left\{\int_I\phi'\left(\frac{1}{c_\theta}\right)du_1du_2-
\int_I\left[\frac{1}{c_\theta}\phi'\left(\frac{1}{c_\theta}\right)-\phi\left(\frac{1}{c_\theta}\right)\right]dC_{n}(u_1,u_2)\right\}
\end{equation}
and
\begin{equation}\label{estimateur 1 param}\arg\sup_{\theta\in\Theta}\left\{\int_I\phi'\left(\frac{1}{c_\theta}\right)~du_1du_2-
\int_I\left[\frac{1}{c_\theta}\phi'\left(\frac{1}{c_\theta}\right)-\phi\left(\frac{1}{c_\theta}\right)\right]~dC_{n}(u_1,u_2)\right\},
\end{equation}
in which $C_{\theta_T}$ is replaced by $C_n$. Note that this class
of estimates contains the maximum pseudo-likelihood (MPL)
estimator proposed by \cite{Oakes1994}; it is obtained for the
$KL_m$-divergence taking $\phi(x)=-\log(x)+x-1$. Under some
regularity conditions, we can prove that these estimates are
consistent and asymptotically normal in the same way as the MPL
estimate when the parameter $\theta_T$ is an interior point of the
parameter space $\Theta.$ The interest of divergence remains in
the fact that a properly choice of the divergence may ameliorate
the MPL one in terms of efficiency-robustness. The results in
\cite{bouzebda-keziou2008} show that, for
$\Theta=[\theta_0,\infty)$, and when the true value $\theta_T$ of
the parameter is equal to $\theta_0$ (corresponding to the
independence assumption), the classical asymptotic normality
property of the MPL estimate is no more satisfied. To overcome
this difficulty,  in what follows, we enlarge the parameter space
$\Theta$ into a wider space $\Theta_e\supset \Theta$. This is
tailored to let $\theta_0$ become an interior point of $\Theta_e$.
More precisely, set
\begin{equation}\label{def de Theta_e }
\Theta_e :=\left\{\theta\in\mathds{R}^d \text{ such that } \int
\left|\phi'(1/c_\theta(u_1,u_2))\right|~du_1du_2 <\infty\right\}.
\end{equation}
So, applying (\ref{eqn 2 dual}), with the class of functions
$$\mathcal{F}:=\left\{(u_1,u_2)\mapsto
\phi'(1/c_\theta(u_1,u_2)); ~\theta\in\Theta_e\right\},$$ we
obtain
\begin{equation}\label{formula 2}
D_\phi(\theta_0,\theta_T)=\sup_{\theta\in\Theta_e}\left\{\int_I\phi'\left(\frac{1}{c_\theta}\right)du_1du_2-
\int_I\left[\frac{1}{c_\theta}\phi'\left(\frac{1}{c_\theta}\right)-\phi\left(\frac{1}{c_\theta}\right)\right]dC_{\theta_T}(u_1,u_2)\right\}.
\end{equation}
Furthermore, the sup in this display is unique and reached in
$\theta=\theta_T.$ Hence, we propose to estimate
$D_\phi(\theta_0,\theta_T)$ by
\begin{equation}\label{estim div}
\widehat{D_\phi}(\theta_0,\theta_T) := \sup_{\theta \in \Theta_e}
\int_{I} m(\theta,u_1,u_2)~dC_{n}(u_1,u_2),
\end{equation}
and to estimate the parameter $\theta_T$ by
\begin{equation}\label{estimate}
\widehat\theta_n := \arg \sup_{\theta \in \Theta_e}
\left\{\int_{I} m(\theta,u_1,u_2)~dC_{n}(u_1,u_2)\right\},
\end{equation}
where
\begin{equation*}
m(\theta,u_1,u_2):=\int_I\phi^{\prime} \left(\frac{1}{c_\theta
(u_1,u_2)}\right)~du_1du_2-
\left\{\phi^{\prime}\left(\frac{1}{c_{\theta}(u_1,u_2)}\right)
 \frac{1}{c_{\theta}(u_1,u_2)}-
 \phi\left(\frac{1}{c_\theta(u_1,u_2)}\right)\right\}.
\end{equation*}
In the sequel we denote by $\frac{\partial}{\partial
\theta}m(\theta,u_1,u_2)$ the $d$-dimensional vector with entries
$\frac{\partial}{\partial\theta_{i}}m(\theta,u_1,u_2)$ and
$\frac{\partial^2}{\partial \theta^2}m(\theta,u_1,u_2)$ the
$d\times d$-matrix with entries
$\frac{\partial^{2}}{\partial\theta_{i}\partial\theta_{j}}m(\theta,u_1,u_2)$.
\begin{remark}Divergences measures have been intensively
used in estimation and test in the framework of the discret
parametric models  with independent identically distributed data;
the estimates of the divergences and the parameter are obtained by
the plug-in method; see \cite{Liese-Vajda1987},
\cite{pardolivre2006} and the references therein. For continuous
parametric models the plug-in procedure does not lead to well
defined estimates; \cite{Keziou2003}, \cite{liesevajda2006},
\cite{Broniatowski-Keziou2008} introduce new estimates and tests,
using the dual representation of divergences, extending the
maximum likelihood procedure.
\end{remark}

\section{The asymptotic behavior of
the estimates}
 \noindent In this section, we provide both weak and strong
consistency of the estimates (\ref{estim div}) and
(\ref{estimate}). We also state their asymptotic normality and
evaluate their limiting variance.
 Statistics of the form
\[\Psi_n :=\int_I\psi(u_1,u_2)~dC_{n}(u_1 ,u_2),\] belong to the general
class of \emph{multivariate rank statistics}. Their asymptotic
properties have been investigated at length by a number of
authors, among whom we may cite \cite{Ruymgaart_Shorack_Zwet1972},
\cite{Ruymgaart1974} and \cite{Ruschendorf1976}. In particular,
the previous authors have provided regularity conditions, imposed
on $\psi(\cdot,\cdot)$, which imply the asymptotic normality of
$\Psi_n$. The corresponding arguments have been modified by
\cite{Genest_Ghoudi_Rivest1995}, as to establish almost sure
convergence of the estimators that they consider (see, e.g.,
\cite{Genest_Ghoudi_Rivest1995} Proposition A.1). In the same
spirit, the limiting behavior, as $n$ tends to the infinity, of
the estimators and test statistics which we will introduce later
on, will make an instrumental use of the general theory of
multivariate rank statistics, and rely, in particular, on
Proposition A.1 in \cite{Genest_Ghoudi_Rivest1995}. The existence
and consistency of our estimators will be established through an
application of the law of the iterated logarithm for empirical
copula processes, in combination with general arguments from
multivariate rank statistics theory (we refer to
\cite{deheuvels1979a}, \cite{fermanianradulovicdragan2004} and
references therein). We will use the following notations
\begin{equation*}
 K_1(\theta,u_1,u_2):=\phi^{\prime}
\left(\frac{1}{c_\theta(u_1,u_2) }\right)
\end{equation*} and
\begin{equation*}
K_2(\theta,u_1,u_2):=\left\{\phi^{\prime}\left(\frac{1}{c_{\theta}(u_1,u_2)}\right)
 \frac{1}{c_{\theta}(u_1,u_2)}-
 \phi\left(\frac{1}{c_{\theta}(u_1,u_2)}\right)\right\}.
 \end{equation*}
\begin{definition}
\begin{enumerate}
\item[(i)] Let $\mathcal{Q}$ be the set of continuous functions
$q$ on $[0,1]$ which are positive on $(0,1)$, symmetric about
$1/2$, increasing on $[0,1/2]$ and satisfy
$\int_0^1\{q(t)\}^{-2}dt< \infty$.
 \item[(ii)] A function
$r: (0,1)\rightarrow (0,\infty)$  is called u-shaped if it is
symmetric about $1/2$ and increasing on $(0,1/2]$. \item[(iii)]
For $0<\beta < 1$ and and u-shaped function $r$, we define
$$
r_\beta(t)=\left\{
\begin{array}{ccc}
  r(\beta t)& if & 0<t\leq 1/2; \\
   r\{1-\beta(1-t)\}& if & 1/2<t\leq 1/2. \\
\end{array}\right.
$$
If for $\beta > 0$ in a neighborhood of $0$, there exists a
constant $M_\beta$, such that $r_\beta \leq M_\beta r$ on $(0,1)$,
then $r$ is called a reproducing u-shaped function. We denote by
$\mathcal{R}$ the set of reproducing u-shaped functions.
\end{enumerate}
\end{definition}
We make use of the following conditions.
\begin{enumerate}
    \item[(C.1)] There exists a neighborhood $N(\theta_T)\subset \Theta_e$ of $\theta_T$
     such that the first and the second partial derivatives with respect to
    $\theta$ of $K_1(\theta,u_1,u_2)$ are dominated on $N(\theta_T)$ by some $\lambda$-integrable
functions;
\end{enumerate}
\begin{enumerate}
    \item[(C.2)] There exists a neighborhood $N(\theta_T)$ of $\theta_T$, such that for all
    $\theta \in N(\theta_T)$, the function $\frac{\partial}{\partial\theta}\,m(\theta,u_1,u_2): (0,1)^2\rightarrow
    \mathds{R}$ is continuously differentiable and there exist functions $r_i \in
    \mathcal{R}$, $\widetilde{r}_i \in \mathcal{R}$
    and $q \in \mathcal{Q}$ ($i,j=1,2,~~i\neq j$ and $\ell,\ell^\prime=1,\ldots,m$) with
    \begin{enumerate}
    \item[(i)]$
    \big|\frac{\partial}{\partial\theta_\ell}\,m(\theta,u_1,u_2)\big|\leq r_1(u_1)r_2(u_2),$
\\$
    \big|\frac{\partial^2}{\partial\theta_\ell\partial u_i}\,m(\theta,u_1,u_2)\big|\leq
    \widetilde{r}_i(u_i)r_j(u_j);$
\item[(ii)] $ \big|m^2(\theta,u_1,u_2)\big|\leq
    \widetilde{r}_i(u_i)r_j(u_j);
$ \item[(iii)] $ \big|\frac{\partial^3}{\partial
\theta^3}K_2(\theta,u_1,u_2)\big|\leq
    \widetilde{r}_i(u_i)r_j(u_j);
$ \item[(iv)]$ \big|m(\theta,u_1,u_2)\big|\leq
    \widetilde{r}_i(u_i)r_j(u_j);
$

\item[(v)]$ \big|\frac{\partial
}{\partial\theta_\ell}m(\theta,u_1,u_2)\big|^2\leq
    \widetilde{r}_i(u_i)r_j(u_j),$\\ $ \big|\frac{\partial^2
}{\partial\theta_\ell\partial\theta_{\ell^\prime}}m(\theta,u_1,u_2)\big|\leq
    \widetilde{r}_i(u_i)r_j(u_j);$
\item[(vi)]$ \big|\frac{\partial}{\partial
u_i}m(\theta,u_1,u_2)\big|\leq
    \widetilde{r}_i(u_i)r_j(u_j)
$
 \end{enumerate}
    and
$ \int_{I}\left\{r_1(u_1)r_2(u_2)\right\}^2
dC_{\theta_T}(u_1,u_2)< \infty, $ $
\int_{I}\left\{q_i(u_i)\widetilde{r}_i(u_i)r_j(u_j)\right\}
dC_{\theta_T}(u_1,u_2)< \infty; $ \item[(C.3)]The matrix $\int
(\partial^2/
\partial^2\theta)
m(\theta,u_1,u_2)dC_{\theta_T}(u_1,u_2)$ is non singular;

\item[(C.4)] The function $(u_1,u_2)\in I\mapsto
\frac{\partial}{\partial\theta}\, m(\theta_T,u_1,u_2)$ is of
bounded variation on $I$.
\end{enumerate}
\begin{theorem}\label{theoreme1}
Assume that conditions (C.1-3) hold.
\begin{enumerate}
\item Let $B(\theta_{T},n^{-1/3}):=\left\{\theta \in \Theta_{e} ,
|\theta -\theta_{T} | \leq n^{-1/3} \right\}$, then as $n$ tends
to infinity, with probability one, the function $\theta\mapsto\int
m(\theta,u_1,u_2)~dC_{n}(u_1,u_2)$ attains its
 maximum value at some point $\widehat\theta_n$ in the interior of  $B(\theta_{T},n^{-1/3})$, which implies that the estimate
 $\widehat\theta_n$ is consistent and satisfies
  \begin{equation*}
  \int \frac{\partial}{\partial \theta}m(\widehat\theta_n,u_1,u_2)~dC_{n}(u_1 ,u_2) = 0.
 \end{equation*}
\item  $\sqrt{n}(\widehat\theta_n -\theta)$ converges in
distribution to a centered multivariate normal random variable
with covariance matrix
\begin{equation}\label{fin}
\mathbf{\Xi}_\phi=\mathbf{S}^{-1}\mathbf{M}\mathbf{S}^{-1},
\end{equation}
with
\begin{equation}\label{S}
\mathbf{S}:=-\int \frac{\partial^2} {\partial \theta^2}
m(\theta_T,u_1,u_2)dC_{\theta_T}(u_1,u_2),
\end{equation}
 and
\begin{equation}\label{M}
\mathbf{M}:=Var\left[\frac{\partial}{\partial
\theta}m(\theta_T,F_1(X_1),F_2(X_2))+W_{1}(\theta_T,X_1) +
W_{2}(\theta_T,X_2)\right],
\end{equation}
where
\begin{equation*}
 W_{i}(\theta_T,X_i) := \int_{I} \mathds{1}_{\{F_{i}(X_i) \leq
 u_i\}}{\textstyle\frac{\partial^2}{\partial\theta \partial
 u_i}}m\left(\theta_T,u_1,u_2\right)
 c_{\theta_T}(u_1,u_2)~du_1du_2,~i=1,2.
\end{equation*}
\end{enumerate}
\end{theorem}
The proof of Theorem \ref{theoreme1} is postponed to section 6.

\section{New tests of independence}
 \noindent One of our
motivation is to build a statistical test of independence, based
on $\phi$-divergence. In the framework of the parametric copula
model, the null hypothesis, i.e., the independence case
$C_{\theta}(u_1,u_2) = u_1u_2$ corresponds to
$\mathcal{H}_0~:~\theta_T = \theta_0$.  We consider the composite
alternative  hypothesis $\mathcal{H}_1~:~\theta_T \neq \theta_0$.
Since, $\theta_0$ is a boundary value of the parameter space
$\Theta$, we can see that the convergence in distribution of the
corresponding pseudo-likelihood ratio statistic to a $\chi^2$
random variable does not hold; see \cite{bouzebda-keziou2008}. We
give now a solution to this problem. We propose the following
statistics
\begin{equation}
\mathbf{T}_n:=\frac{2n}{\phi^{\prime
\prime}(1)}\widehat{D_\phi}(\theta_0,\theta_T).
\end{equation}
We will see that the proposed statistic converges in distribution,
under the null hypothesis $\mathcal{H}_0$, to a $\chi^2$ random
variable with $d$ degrees of freedom, which permits to build a
test of $\mathcal{H}_0$ against $\mathcal{H}_1$ asymptotically of
level $\alpha$. The limit law of $\mathbf{T}_n$ is given also
under the alternative hypothesis $\mathcal{H}_1$. We will use the
following additional conditions
\begin{enumerate}
\item [(C.5)] We have
$$\lim_{\theta\rightarrow \theta_0 }\frac{\partial^2}{\partial\theta_\ell
\partial u_i}\,m(\theta,u_1,u_2)=0,$$ and there exist $M_1>0$ and
$\delta_1>0$ such that, for all $\theta$ in some neighborhood of
$\theta_0$, one has, for $i=1,2$,
\begin{equation*}
\big|\frac{\partial^2}{\partial\theta_\ell\partial
u_i}\,m(\theta,u_1,u_2)c_{\theta_T}(u_1,u_2)\big|<
M_1r(u_i)^{-1.5+\delta_1}r(u_{3-i})^{0.5+\delta_1},
\end{equation*}
where $r(u) := u(1-u)$ for $u\in
    (0,1)$.
 \end{enumerate}
 \begin{remark}
When $\theta_T=\theta_0$, under the conditions (C.1) and (C.5) we
can see that $\mathbf{S}$ and $\mathbf{M}$ can be written as
$$\mathbf{S}=\mathbf{M}=\int \left[\frac{\partial} {\partial \theta}
m(\theta_T,u_1,u_2)\right]\left[\frac{\partial} {\partial \theta}
m(\theta_T,u_1,u_2)\right]^\top dC_{\theta_T}(u_1,u_2).$$
\end{remark}
\begin{theorem}\label{theoreme2}

\begin{enumerate}
\item [(1)] Assume that conditions (C.1-5) hold. If
$\theta_T=\theta_{0}$, then the statistic $\mathbf{T}_n$ converges
in distribution to a $\chi^{2}$ variable with $d$ degrees of
freedom. \item [(2)] Assume that conditions (C.1-4) hold. If
$\theta_T\neq \theta_{0}$, then
$\sqrt{n}\left(\widehat{D_\phi}(\theta_0,\theta_{T})-D_\phi(\theta_0
,\theta_{T})\right)$ converges in distribution to a centered
normal variable with variance
\begin{equation}\label{mm}
\sigma^2_{\phi}(\theta_0,\theta_T) :=
Var\left[m(\theta_T,F_1(X_1),F_2(X_2))+Y_{1}(\theta_T,X_1) +
Y_{2}(\theta_T,X_2)\right],
\end{equation}
\end{enumerate}
where
\begin{equation*}
 Y_{i}(\theta_T,X_i) := \int_{I} \mathds{1}_{\{F_{i}(X_i) \leq
 u_i\}}{\textstyle\frac{\partial}{ \partial
 u_i}}m\left(\theta_T,u_1,u_2\right)
 c_{\theta_T}(u_1,u_2)~du_1du_2,~i=1,2.
\end{equation*}
\end{theorem}
The proof of Theorem \ref{theoreme2} is postponed to section 6.
\begin{remark} An application of Theorem \ref{theoreme2}, leads to reject the
null hypothesis $\mathcal{H}_0:\theta_T=\theta_0$, whenever the
value of the statistic ${\bf T}_n$ exceeds $q_{1-\alpha}$, namely,
the $(1-\alpha)$-quantile of the $\chi^2$ law with $d$ degrees of
freedom. The corresponding test is then, asymptotically of level
$\alpha$, when $n\rightarrow\infty$. The critical region is,
accordingly, given by
\begin{equation*}
CR := \left\{{\bf T}_n > q_{1-\alpha}\right\}.
\end{equation*}
The fact that this test is consistent follows from Theorem
\ref{theoreme2}. Further, this theorem can be used to give an
approximation to the power function $\theta_T\in\Theta\mapsto
\beta(\theta_T):=P_{\theta_T}\left\{CR\right\}$ in a similar way
to  \cite{KeziouLeoni2007}. We so obtain that
\begin{equation}\label{marre2}
\beta(\theta_T)\approx 1-\Phi\left(
\frac{\sqrt{n}}{\sigma_{\phi}(\theta_0,\theta_T)}
\left(\frac{q_{1-\alpha}}{2n}-D_\phi(\theta_0
,\theta_{T})\right)\right),
\end{equation}
where $\Phi$ denotes, as usual, the cumulative distribution
function of a $N(0,1)$ standard normal random variable. A useful
consequence of (\ref{marre2}) is the possibility of computing an
approximate value of the sample size ensuring a specified power
$\beta(\theta_T)$, with respect to some pre-assigned alternative
$\theta_T\neq\theta_0.$ Let $n_0$ be the positive root of the
equation
\begin{equation*}
\beta = 1-\Phi\left(
\frac{\sqrt{n}}{\sigma_{\phi}(\theta_0,\theta_T)}
\left(\frac{q_{1-\alpha}}{2n}-D_\phi(\theta_0
,\theta_{T})\right)\right),
\end{equation*}
which can be rewritten into
\begin{equation*}
n_0=\frac{(a+b)- \sqrt{a(a+2b)}}{2D_\phi(\theta_0 ,\theta_{T})^2},
\end{equation*}
where $a
:=\sigma_{\phi}(\theta_0,\theta_T)\left(\Phi^{-1}(1-\beta)\right)^2$
and $b:=q_{1-\alpha} D_\phi(\theta_0 ,\theta_{T})$. The
sought-after approximate value of the sample size is then given
$$n^*:=\lfloor n_0\rfloor+1,$$
where $\lfloor u\rfloor$ denote the integer part of $u$.
\end{remark}
\begin{remark}
From Theorem \ref{theoreme1} and \ref{theoreme2}, it is clear that
an asymptotic $1-\alpha$ confidence interval or region,
$\mathcal{R}_\alpha$ about $\theta$ can be easily constructed
using the Intersection method as described in \cite{feng1992}.
\end{remark}

\begin{remark}  The above regularity conditions are satisfied by
a large number of  parametric families of bivariate copulas; see
for instance
 \cite{tsukahara2005}.
\end{remark}
\begin{remark}
The parameters   (\ref{S}) and (\ref{M})   may be consistently
estimated respectively  by the sample mean of
\begin{equation}\label{S1}
 \frac{\partial^2}{\partial \theta^2}
m(\widehat{\theta}_n,F_{1n}(X_{1,k}),F_{2n}(X_{2,k})), \quad
k=1,\ldots,n,
\end{equation}
and the sample variance of
\begin{equation}
 {\textstyle\frac{\partial}{\partial\theta}}\,m\left(\widehat{\theta}_n,F_{1n}
 (X_{1,k}),F_{2n}(X_{2,k})\right) + W_{1}(\widehat{\theta}_n,X_{1,k}) +
 W_{2}(\widehat{\theta}_n,X_{2,k}), \quad k=1,\ldots,n,
\end{equation}
as was done in \cite{Genest_Ghoudi_Rivest1995}. The asymptotic
variance (\ref{mm}) can be consistently estimated in the same way.
\end{remark}
\section{Concluding remarks}
We have introduced a new estimation and test procedure in
parametric copula models with unknown margins. The methods is
based on divergences between copulas and the duality technique. It
generalizes the maximum pseudo-likelihood one, and applies both
when the parameter is an interior or a boundary value, in
particular for testing the null hypothesis of independence. It
will be interesting to investigate the problem of the choice of
the divergence which leads to an ``optimal'' (in some sense)
estimate or test in terms of efficiency and robustness.
\section{Appendix}\label{appendix} \noindent First we give a technical Lemma which
we will use to prove our results.
\begin{lemma}\label{lemma1}
Let $F_{\theta_T}$ have a continuous margins and let
$C_{\theta_T}$ have continuous partial derivatives. Assume that
$\jmath$ is a continuous function, with bounded variation. Then
\begin{equation}
\int_{I} \jmath(u_1,u_2)~d\left(C_{n}(u_1,u_2)- C(u_1,u_2)\right)=
O\left(n^{-1/2}(\log\log n)^{1/2}\right)~~ (a.s.).
\end{equation}
\end{lemma}

\noindent{\bf{Proof of Lemma \ref{lemma1}.}}  Recall that the
\emph{modified empirical copula} $C_n$, is slightly different from
the \emph{empirical copula} $\mathds{C}_n(u_1,u_2)$, introduced by
\cite{deheuvels1979a}, and defined by
\begin{equation}\label{marre3}\mathds{C}_n(u_1,u_2)=F_n\Big(F_{1n}^{-1}(u_1),
F_{2n}^{-1}(u_2)\Big)\quad\hbox{for}\quad(u_{1},u_{2})\in
(0,1)^2,\end{equation} where $F_{1n}^{-1}(\cdot)$ and
$F_{2n}^{-1}(\cdot)$ denote the empirical quantile functions,
associated with $F_{1n}(x_1)=F_n(x_1,\infty)$ and
$F_{2n}(x_2)=F_n(\infty,x_2)$, respectively, and defined by
\begin{equation}
F_{jn}^{-1}(t):=\inf\{x\in \mathds{R}\mid F_{jn}(x)\geq t\},\quad
j=1,2.
\end{equation}
Here,  $F_n(\cdot,\cdot)$ denotes the joint empirical distribution
function, associated with the sample $\{(X_{1k},X_{2k});~
k=1,\ldots,n\},$  defined by
\begin{equation}
F_{n}(x_1,x_2)=\frac{1}{n} \sum _{k=1}^{n}
\mathds{1}_{\left\{X_{1k}\leq
x_1\right\}}\mathds{1}_{\left\{X_{2k}\leq x_2\right\}},~ -\infty <
x_1,x_2< \infty.
\end{equation}
We know that $\mathds{C}_n$ and $C_n$ coincide on the grid
$\left\{\left(i/n,j/n\right),~1\leq i\leq j \leq n \right\}.$ The
subtle difference lies in the fact that $\mathds{C}_n$ is
left-continuous with right-hand limits, whereas $C_n$ on the other
hand is right continuous with left-hand limits. The difference
between $\mathds{C}_n$ and $C_n$, however, is small
\begin{equation}\label{app 1}
\sup_{{\bf{u}}\in
 I}\left|{\mathds{C}}_{n}({\bf{u}})-C_{n}({\bf{u}})\right|\leq
\max_{1\leq i,j\leq
n}\left|{\mathds{C}}_{n}\left(\frac{i}{n},\frac{j}{n}\right)-\mathds{C}_{n}
\left(\frac{i-1}{n},\frac{j-1}{n}\right)\right|\leq \frac{2}{n}.
\end{equation}
Using integration by parts, as in
\cite{fermanianradulovicdragan2004}, we see that
\begin{eqnarray}
\lefteqn{\sqrt{n}\int_{I}\jmath(u_1,u_2)~d(C_{n}- C)(u_1,u_2) =
\int_{I}\sqrt{n}(C_n- C)(u_1,u_2)~d\jmath(u_1,u_2)}\nonumber\\
& &-  \int_{I} \sqrt{n}(C_n- C)(u_1,1)~d\jmath(u_1,u_2)- \int_{I}
\sqrt{n}(C_n- C)(1,u_2)~d\jmath(u_1,u_2)\nonumber\\
& &-  \int_{[0,1]} \sqrt{n}(C_n(u_1,1)-u_1)~d\jmath(u_1,1)-
\int_{[0,1]} \sqrt{n}(C_n(1,u_2)-u_2)~d\jmath(1,u_2).\nonumber
\end{eqnarray}
Hence,
\begin{equation*}
\left|\sqrt{n}\int_{I}\jmath(u_1,u_2)~d(C_{n}-C)(u_1,u_2)\right|\leq
5\sqrt{n}\sup_{{\bf u}\in I}\left|(C_{n} - C)(\bf
u)\right|\int_{I}d\left|\jmath(\bf u)\right|.
\end{equation*}
From this and (\ref{app 1}), applying Theorem 3.1 in
\cite{deheuvels1979a}, we obtain the following result
$$\int_{I} \jmath(u_1,u_2)~d(C_{n}-C)(u_1,u_2)=
O\left(n^{-1/2}(\log\log n)^{1/2}\right)~~ (a.s.).$$ \hfill
$\blacksquare$\\ \noindent\textbf{Proof of Theorem
\ref{theoreme1}} (1) Under the Assumption (C.1) and (C.2.iii), a
simple calculation gives
\begin{equation}\label{5.3}
\int_I\frac{\partial}{\partial \theta}m(\theta ,u_1, u_2)
dC_{\theta_T}(u_1, u_2) = 0,
\end{equation}
and
\begin{equation}\label{vv}
\int_I\frac{\partial^2}{\partial \theta^2}m(\theta ,u_1, u_2)
dC_{\theta_T}(u_1, u_2)=-\int\phi
^{\prime\prime}\left(\frac{1}{c_{\theta_T}}\right)\frac{{\dot{c}_{\theta_T}\dot{c}_{\theta_T
}^{\top}}}{c_{\theta_T}^3}~d\lambda=-\mathbf{S}
\end{equation}
We see that the matrix $\mathbf{S}$ is symmetric and positive
using the fact that the second derivative $\phi^{''}(\cdot)$ is
nonnegative by the assumption that the function $\phi(\cdot)$ is
convexe. Hence, $\mathbf{S}$ is positive definite by (C.3).
Introduce the statistic $\mathbf{\Phi}_n(\theta_T)$ defined by
\begin{equation}
\mathbf{\Phi}_n(\theta_T):=   \int_{I} \frac{\partial}{\partial
\theta}m(\theta,u_1,u_2)~dC_{n}(u_1,u_2),
\end{equation}
and combine (\ref{5.3}) with Theorem 2.1 in
\cite{Ruymgaart_Shorack_Zwet1972} to show that, as $n\rightarrow
\infty $
\begin{equation}\label{nom}
\sqrt{n}\mathbf{\Phi}_n(\theta_T)\stackrel{d}{\rightarrow}
N(0,\mathbf{M}),
\end{equation}
where $\mathbf{M}$ is defined in (\ref{M}). Denote
\begin{equation}\label{vas}
\mathbf{\Upsilon}_n(\theta_T):=\int_{I} \frac{\partial^2}{\partial
\theta^2}m(\theta,u_1,u_2)~dC_{n}(u_1,u_2),
\end{equation}
we make use of (\ref{vv}) and (C.2.v) in combinaison with
Proposition A.1 of \cite{Genest_Ghoudi_Rivest1995}, one finds
\begin{equation}\label{saa}
\mathbf{\Upsilon}_n(\theta_T)\rightarrow -\mathbf{S},~~ (a.s.).
\end{equation}
We recall that $\mathbf{S}$ is in (\ref{S}). Now, for any $\theta
= \theta_T + v n^{-1/3}$ with $|v| \leq 1$, consider a Taylor
expansion of  $\int m(\theta,u_1,u_2)~dC_{n}(u_1,u_2)$ in $\theta$
around $\theta_T$, and use (\ref{saa}), and (C.2.iii) to obtain
\begin{eqnarray}\label{dev 1}
\nonumber &&n\int_{I} m(\theta,u_1,u_2)dC_{n}(u_1,u_2)
-n\int_{I} m(\theta_{T},u_1,u_2)~dC_{n}(u_1,u_2)=\\
 & &
n^{2/3}v^\top\mathbf{\Phi}_n(\theta_T)+2^{-1}n^{1/3}v\mathbf{S}v^\top+O(1)~~(a.s.)
\end{eqnarray}
uniformly in $v$ with $|v|\leq 1$. On the other hand, since $\int
\frac{\partial}{\partial\theta_\ell}m(\theta_T,u_1,u_2)^{2}~dC_{\theta_T}(u_1,u_2)<
\infty$, and
$\frac{\partial}{\partial\theta_\ell}m(\theta,\cdot,\cdot)$ is of
bounded variation by assumption (C.4)($\ell=1,\ldots,d$), using
Lemma \ref{lemma1} we can show that
\begin{equation}\label{22}
\int_{I}
\frac{\partial}{\partial\theta_\ell}m(\theta_T,u_1,u_2)~dC_{n}(u_1,u_2)=
O\left(n^{-1/2}(\log\log n)^{1/2}\right)~~ (a.s.).
\end{equation}
  Therefore,
using (\ref{dev 1}) and (\ref{22}), we obtain for any $\theta =
\theta_T + v n^{-1/3}$  with $|v|=1$:
\begin{eqnarray}\label{vie}
\nonumber&&n\int_{I} m(\theta,u_1,u_2)~dC_{n}(u_1,u_2) -n\int_{I}
m(\theta_{T},u_1,u_2)~dC_{n}(u_1,u_2) \\&& \leq O(n^{1/6}(\log\log
n)^{1/2})- 2^{-1}\vartheta n^{1/3} +O(1) ~~(a.s.),
\end{eqnarray}
where $\vartheta$ is the smallest eigenvalue of the matrix
$\mathbf{S}$. Observe  that $\vartheta$ is positive since
$\mathbf{S}$ is symmetric, positive and non singular by assumption
(C.3) Using (\ref{vie}) and the fact that the function $\theta
\mapsto \int_{I} m(\theta_T,u_1,u_2)~dC_{n}(u_1,u_2)$ is
continuous, we conclude that as $n \rightarrow \infty$, with
probability one, $\theta \mapsto \int_{I}
m(\theta_T,u_1,u_2)~dC_{n}(u_1,u_2)$ reaches its maximum value at
some point $\widehat \theta_n$ fulfills $\int_{I}
\frac{\partial}{\partial\theta}m(\widehat\theta_n,u_1,u_2)~dC_{n}(u_1,u_2)=0$
 and $|\widehat\theta_n-\theta_T|=O(n^{-1/3})$.

\noindent(2) Using the first part of Theorem \ref{theoreme1}, by a
Taylor expansion of
$$\int_{I}
\frac{\partial}{\partial\theta}m(\widehat\theta_n,u_1,u_2)~dC_{n}(u_1,u_2)$$
in $\widehat\theta_n$ around $\theta_T$, we obtain
\begin{eqnarray*}
\lefteqn{0 = \int_{I} \frac{\partial}{\partial\theta}
m(\widehat\theta_n,u_1,u_2)~dC_{n}(u_1,u_2)}\\
& = & \int_{I}
\frac{\partial}{\partial\theta}m(\theta_T,u_1,u_2)~dC_{n}(u_1,u_2)
 + (\widehat\theta_n - \theta_{T})^\top\int_{I}
\frac{\partial^2}{\partial\theta^2}(\theta_T,u_1,u_2)~dC_{n}(u_1,u_2)+o(n^{-1/2}).
\end{eqnarray*}
Hence,
\begin{equation}\label{equivaassmpto}
\sqrt{n}(\widehat\theta_n - \theta_{T}) =-\left[
\mathbf{\Upsilon}_n(\theta_T)\right]^{-1}\sqrt{n}\mathbf{\Phi}_n(\theta_T)
+o_P(1)
\end{equation}
Using (\ref{nom}) and (\ref{saa}), by Slutsky theorem, we conclude
then
\begin{equation}
\sqrt{n}(\widehat\theta_n - \theta_{T}) \rightarrow
N(0,\mathbf{\Xi}_\phi),
\end{equation}
where we recall that $\mathbf{\Xi}_\phi$ is defined in
(\ref{fin}).\hfill$\blacksquare$  \vskip5pt

\noindent\textbf{Proof of Theorem \ref{theoreme2}}\\\noindent(1)
Assume that $\theta_T=\theta_0$. Hence, from
(\ref{equivaassmpto}), using (\ref{vv}), we obtain
\begin{equation}\label{eqn 1}
\sqrt{n}\left(\widehat{\theta}_n-\theta_T\right)=-\mathbf{S}^{-1}
\sqrt{n}\mathbf{\Phi}_n(\theta_T)+o_P(1).
\end{equation}
On the other hand, expanding in Taylor series
$$\frac{2n}{\phi^{\prime
\prime}(1)}\widehat{D_\phi}(\theta_0,\widehat\theta_n)=\frac{2n}{\phi^{\prime
\prime}(1)}\int_{I} m(\widehat\theta_n,u_1,u_2)~dC_{n}(u_1,u_2)$$
in $\widehat\theta_n$ around $\theta_T$, in combination with the
fact that $\int_{I} m(\theta_T,u_1,u_2)~dC_{n}(u_1,u_2)=0$, we get
\begin{equation}
\frac{2n}{\phi^{\prime
\prime}(1)}\widehat{D_\phi}(\theta_0,\widehat\theta_n)=\frac{2n}{\phi^{\prime
\prime}(1)}\mathbf{\Phi}_n(\theta_T)(\widehat{\theta}_n-\theta_T)-
\frac{n}{\phi^{\prime
\prime}(1)}(\widehat{\theta}_n-\theta_T)^\top\mathbf{\Upsilon}_n(\theta_T)
(\widehat{\theta}_n-\theta_T)+o_P(1).
\end{equation}
Using (\ref{vv}), (\ref{eqn 1}) and the fact that
$\mathbf{S}=\phi^{''}(1)I_{\theta_T}$ ($I_{\theta_T}$ denotes the
Fisher information matrix) when $\theta_T=\theta_0$ to obtain
\begin{equation}
\frac{2n}{\phi^{\prime
\prime}(1)}\widehat{D_\phi}(\theta_0,\widehat\theta_n)=\frac{1}{\phi^{\prime
\prime}(1)}\sqrt{n}\mathbf{\Phi}_n(\theta_T)I_{\theta_T}^{-1}\sqrt{n}\mathbf{\Phi}_n(\theta_T).
\end{equation}
Finally, use the convergence in (\ref{nom}) and the fact that
$\mathbf{M}=\phi^{''}(1)I_{\theta_T}$  when  $\theta_T=\theta_0$,
to conclude that $[\frac{2n}{\phi^{\prime
\prime}(1)}]\widehat{D_\phi}(\theta_0,\theta_T)$ converge in
distribution to a $\chi^2$ variable with $d$ degrees of freedom
when $\theta_T=\theta_0$. \hfill$\blacksquare$  \vskip5pt

\noindent (2) Assume that $\theta_T\neq \theta_0$, using Taylor
expansion again of $$\widehat{D_\phi}(\theta_T,\theta_0)=\int_{I}
m(\widehat\theta_n,u_1,u_2)~dC_{n}(u_1,u_2)$$ in
$\widehat{\theta}_n$ around $\theta_T$, combined with the fact
that $\int_{I} \frac{\partial}{\partial\theta}
m(\theta_T,u_1,u_2)~dC_{\theta_T}(u_1,u_2)=0$, we obtain from part
(2) of Theorem \ref{theoreme1}
$$\int_{I} m(\widehat\theta_n,u_1,u_2)~dC_{n}(u_1,u_2)=\int_{I} m(\theta_T,u_1,u_2)~dC_{n}(u_1,u_2)+o_P(n^{-1/2}).$$
Hence,
\begin{eqnarray*}
\lefteqn{\sqrt{n}\left(\widehat{
D_\phi}(\theta_0,\theta_T)-D_\phi(\theta_0,\theta_T)\right)=}\\
&&\sqrt{n}\left(\int_{I}
m(\theta_T,u_1,u_2)~dC_{n}(u_1,u_2)-\int_{I}
m(\theta_T,u_1,u_2)~dC_{\theta_T}(u_1,u_2)\right)+o_P(1),
\end{eqnarray*}
which under assumption (C.2.ii) and (C.2.vi) by  Theorem 2.1 in
\cite{Ruymgaart_Shorack_Zwet1972} once more, converges to a
centred normal variable with variance given in (\ref{mm}).
\hfill$\blacksquare$  \vskip5pt

\noindent \textbf{Acknowledgement.} The authors thank Professors
Michel Broniatowski and Paul Deheuvels  for their helpful
discussions and suggestions leading to improvement of this paper.


TITLE: $\lim_{x \rightarrow \infty}$ : problem with method I'm using
QUESTION [1 upvotes]: $\lim_{x \rightarrow \infty}  \sqrt{x^2+7x}-x$
I'm having trouble finding the problem with the method I'm using:
$\lim_{x \rightarrow \infty}  \sqrt{x^2(1+\frac{7}{x})}-x$
$\lim_{x \rightarrow \infty}  x\sqrt{1+\frac{7}{x}}-x$
$\lim_{x \rightarrow \infty}  \frac{7}{x}=0$
$\lim_{x \rightarrow \infty}  x\sqrt{1+0}-x$
so I get 0 for my answer. I know that graphing the function makes it clear that this is not the case but I don't see whats wrong with what I did above.

REPLY [1 votes]: Idea to solve the problem using the expression's conjugate:
$$\left(\sqrt{x^2+7x}-x\right)\cdot\frac{\left(\sqrt{x^2+7x}+x\right)}{\left(\sqrt{x^2+7x}+x\right)}=\frac{x^2+7x-x^2}{\sqrt{x^2+7x}+x}=$$
$$=\frac 7{\sqrt{1+\frac7x}+1}\xrightarrow[x\to\infty]{}\frac72$$
The problems with what you did: (1) You took the limit only of part of the expression ($\frac7x$) while leaving the rest untouched, and this is strictily forbidden unless you prove otherwise in your particular case, and (2) even after this, you remain with an expression that doesn't help you solve the problem as you get " x time something minus x"... and that can be anything.

REPLY [1 votes]: Hint: When you have that $$\lim_{x \rightarrow \infty} f(x) = \infty$$ and $$\lim_{x \rightarrow \infty} g(x) = \infty,$$ can you then conclude that $$\lim_{x \rightarrow \infty} [f(x)-g(x)] = 0?$$
(Consider $f(x) = 2x$ and $g(x) = x$.)


TITLE: The normalizer of a torus is a closed subgroup of the Lie group
QUESTION [1 upvotes]: Let $G$ be a Lie group and $T \subset G$ a torus, show that $N(T)$ is a closed subgroup of $G$.  Could somebody give a sketch of the proof?

REPLY [1 votes]: Say $n_i\in N(T)$ converges to $n$, and let $t\in T$.
Then $$n_itn_i^{-1}\rightarrow ntn^{-1}.$$
Since $n_itn_i^{-1}\in T$ we have $ntn^{-1}\in T$ since $T$ is closed.


text:
\section{Components are Open iff Union of Open Connected Sets/Components are Open implies Space is Union of Open Connected Sets}
Tags: Connected Spaces, Components

\begin{theorem}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let the [[Definition:Component (Topology)|components]] of $T$ be [[Definition:Open Set (Topology)|open sets]].
Then:
:$S$ is a [[Definition:Set Union|union]] of [[Definition:Open Set (Topology)|open]] [[Definition:Connected Set (Topology)|connected sets]] of $T$.
\end{theorem}

\begin{proof}
Let the [[Definition:Component (Topology)|components]] of $T$ be [[Definition:Open Set (Topology)|open]].
By definition, the [[Definition:Component (Topology)|components]] of $T$ are a [[Definition:Set Partition|partition]] of $S$.
Hence $S$ is the [[Definition:Set Union|union]] of the [[Definition:Open Set (Topology)|open]] [[Definition:Component (Topology)|components]] of $T$.
Since a [[Definition:Component (Topology)|component]] is a [[Definition:Maximal Set|maximal]] [[Definition:Connected Set (Topology)|connected set]] by definition, then $S$ is a [[Definition:Set Union|union]] of [[Definition:Open Set (Topology)|open]] [[Definition:Connected Set (Topology)|connected sets]] of $T$.
\end{proof}

config: wiki


text:
\begin{document}

\maketitle

\begin{abstract}
We derive two-sided bounds for expected values of suprema of canonical processes based on random variables
with moments growing regularly. We also discuss a Sudakov-type minoration principle for canonical
processes.
\end{abstract}

\section{Introduction and Main Results}

In many problems arising in probability theory and its applications one needs to estimate the supremum of a stochastic
process. In particular it is very useful to be able to find two-sided bounds for the mean of the
supremum. The modern approach to this challenge is based on the chaining methods, see monograph \cite{Talnew}.

In this note we study the class of so-called \emph{canonical processes} of the form
$X_t=\sum_{i=1}^{\infty}t_iX_i$, where $X_i$ are independent random variables. If $X_i$ are \emph{standardized},
i.e. have mean zero and variance one, then the above series converges a.s.\  for $t\in \ell^2$ and we may try
to estimate $\Ex\sup_{t\in T}X_t$ for $T\subset \ell^2$.
To avoid measurability questions we either assume that the index set $T$ is countable or define
in a general situation
\[
\Ex\sup_{t\in T}X_t:=\sup\left\{\Ex\sup_{t\in F}X_t\colon\ F\subset T \mbox{ finite }\right\}.
\]
It is also more convenient to work with the quantity  $\Ex\sup_{s,t\in T}(X_t-X_s)$ rather than $\Ex\sup_{t\in T}X_t$.
Observe however that if the set $T$ or the variables $X_i$ are symmetric then
\[
\Ex\sup_{s,t\in T}(X_s-X_t)=\Ex\sup_{s\in T}X_s+\Ex\sup_{t\in T}(-X_t)=2\Ex\sup_{t\in T}X_t.
\]

In the case when $X_i$ are i.i.d.\ ${\mathcal N}(0,1)$ r.v.s, $X_t$ is the canonical Gaussian process. Moreover,  any
centered separable Gaussian process has the Karhunen-Lo\`eve representation of such form (see e.g. Corollary 5.3.4 in \cite{Marcus}). In the Gaussian case
the behaviour of the supremum
of the process is related to the geometry of the metric space $(T,d_2)$, where $d_2$ is the $\ell^2$-metric
$d(s,t)=(\Ex|X_s-X_t|^2)^{1/2}$. The celebrated Fernique-Talagrand majorizing measure bound
(cf. \cite{Fe,Ta_reg}) may be expressed in the form
\[
\frac{1}{C}\gamma_2(T)\leq \Ex\sup_{t\in T}X_t\leq C\gamma_2(T),
\]
where here and in the sequel $C$ denotes a universal constant,
\[
\gamma_2(T):=\inf\sup_{t\in T}\sum_{n=0}^{\infty}2^{n/2}\Delta_2(A_n(t)),
\]
the infimum runs over all admissible sequences of partitions $({\mathcal A}_n)_{n\geq 0}$ of the
set $T$, $A_n(t)$ is the unique set in ${\mathcal A}_n$ which contains $t$, and $\Delta_2$ denotes the $\ell^2$-diameter.
An increasing sequence of partitions $({\mathcal A}_n)_{n\geq 0}$ of $T$ is called \emph{admissible} if ${\cal A}_0=\{T\}$
and $|{\cal A}_n|\leq N_n:=2^{2^n}$ for $n\geq 1$.

In \cite{Ta_can} Talagrand derived two-sided bounds for suprema of the canonical processes based on i.i.d.\ symmetric
r.v.s $X_i$ such that $\Pr(|X_i|> t)=\exp(-|t|^p)$, $1\leq p<\infty$. This result was later extended in \cite{La_sup}
to the case of variables with (not too rapidly decreasing) log-concave tails, i.e.\ to the case when
$X_i$ are symmetric, independent, $\Pr(|X_i|\geq t)=\exp(-N_i(t))$, $N_i\colon [0,\infty)\rightarrow [0,\infty)$
are convex and $N_i(2t)\leq \gamma N_i(t)$ for $t>0$ and some constant $\gamma$.
The aim of this note is to find two-sided bounds for suprema
for a more general class of canonical processes.

For a general process $(X_t)_{t\in T}$ one needs to study a family of metrics instead of a single one. We
define
\[
d_p(s,t):=\|X_s-X_t\|_p,\quad p\geq 1,\ s,t\in T,
\]
where for a real random variable $Y$ and $p\geq 1$, $\|Y\|_p:=(\Ex|Y|^p)^{1/p}$ denotes the $L^p$-norm of $Y$.
Following ideas of Talagrand, we define the functional
\[
\gamma_X(T):=\inf\sup_{t\in T}\sum_{n=0}^{\infty}\Delta_{2^n}(A_n(t)),
\]
where $\Delta_{p}$ denotes the diameter with respect to the  distance $d_p$ and as in the case of the $\gamma_2$-functional
the infimum runs over all admissible sequences of partitions $({\cal A}_n)$ of the set $T$.

It is not hard to show (as it was noted independently by Mendelson and the first named author, c.f.
\cite[Exercise 2.2.25]{Talnew}) that for \emph{any}
process $(X_t)_{t\in T}$,
\begin{equation}
\label{eq:upper}
\Ex\sup_{s,t\in T} (X_s-X_t)\leq C\gamma_X(T).
\end{equation}

To reverse bound \eqref{eq:upper} we need some regularity assumptions. We express them for canonical processes
in terms of moments growth of variables $X_i$. It is easy to check that
for a symmetric variable $Y$ with a log-concave tail $\exp(-N(t))$, $\|Y\|_p\leq C\frac{p}{q}\|Y\|_q$ for $p\geq q\geq 2$.
Moreover, the additional condition
$N(2t)\leq \gamma N(t)$ yields $\|Y\|_{\beta p}\geq 2 \|Y\|_p$ for $p\geq 2$ and a constant $\beta$ which depends only
on $\gamma$.
This motivates the following definitions.

\begin{dfn}
For $\alpha\geq 1$ we say that moments of a random variable $X$ \emph{grow $\alpha$-regularly} if
\[
\|X\|_p\leq \alpha\frac{p}{q}\|X\|_q\quad \mbox{ for }p\geq q\geq 2.
\]
\end{dfn}

\begin{dfn}
For $\beta<\infty$ we say that moments of a random variable $X$ \emph{grow with speed $\beta$} if
\[
\|X\|_{\beta p}\geq 2\|X\|_p \quad \mbox{ for }p\geq 2.
\]
\end{dfn}

The class of all standardized random variables with the $\alpha$-regular growth of moments
will be denoted by ${\mathcal R}_{\alpha}$ and with moments growing with speed $\beta$ by
${\mathcal S}_{\beta}$.

\begin{thm}
\label{thm:supcan}
Let $X_t=\sum_{i=1}^{\infty}t_iX_i$, $t\in \ell^2$ be the canonical process based on independent standardized
r.v.s $X_i$ with  moments growing $\alpha$-regularly with
speed  $\beta$ for some $\alpha\geq 1$ and $\beta>1$. Then for any $T\subset \ell^2$,
\[
\frac{1}{C(\alpha,\beta)}\gamma_X(T)\leq \Ex\sup_{s,t\in T}(X_s-X_t)\leq C\gamma_X(T).
\]
\end{thm}

Here and in the sequel $C(\alpha,\beta)$ denotes a constant which depends only on $\alpha$ and $\beta$
(which may differ at each occurrence).
The above result easily yields the following comparison result for suprema of processes.

\begin{cor}\label{cor:compmom}
Let $X_t$ be as in Theorem \ref{thm:supcan}. Then for any nonempty  $T\subset \ell^2$ and any process $(Y_t)_{t\in T}$
such that $\|Y_s-Y_t\|_p\leq \|X_s-X_t\|_p$ for $p\geq 1$ and $s,t\in T$ we have
\[
\Ex\sup_{s,t\in T}(Y_s-Y_t)\leq C(\alpha,\beta)\Ex\sup_{s,t\in T}(X_s-X_t).
\]
\end{cor}

\begin{proof}
The assumption implies $\gamma_Y(T)\leq \gamma_X(T)$ and the result immediately follows by the lower bound in Theorem
\ref{thm:supcan} and estimate \eqref{eq:upper} used for the process $Y$.
\end{proof}

In fact one may show a stronger result.
\begin{cor}
\label{cor:comptails}
Let $X_t$ and $Y_t$ be as in Corollary \ref{cor:compmom}. Then for $u\geq 0$,
\[
\Pr\left(\sup_{s,t\in T}(Y_s-Y_t)\geq u\right)\leq
C(\alpha,\beta)\Pr\left(\sup_{s,t\in T}(X_s-X_t)\geq \frac{1}{C(\alpha,\beta)}u\right).
\]
\end{cor}

Another consequence of Theorem \ref{thm:supcan} is the following striking bound for suprema of some canonical processes.

\begin{cor}
\label{cor:convhull}
Let $X_t$ be as in Theorem \ref{thm:supcan} and $T\subset \ell^2$ be such that $\Ex\sup_{s,t\in T}(X_s-X_t)<\infty$.
Then there exist $t^1,t^2,\ldots\in \ell^2$ such that $T-T\subset\overline{\mathrm{conv}}\{\pm t^n\colon\ n\geq 1\}$
and $\|X_{t^n}\|_{\log(n+2)}\leq C(\alpha,\beta)\Ex\sup_{s,t\in T}(X_s-X_t)$.
\end{cor}

\begin{rem}
The reverse statement easily follows by the union bound and Chebyshev's inequality. Namely, for any canonical process
$(X_t)_{t\in \ell^2}$ and any nonempty set $T\subset \ell^2$ such that
$T-T\subset\overline{\mathrm{conv}}\{\pm t^n\colon\ n\geq 1\}$
and $\|X_{t^n}\|_{\log(n+2)}\leq M$ one has $\Ex\sup_{s,t\in T}(X_s-X_t)\leq CM$. For details see the argument after
Corollary 1.2 in \cite{BL}.
\end{rem}

\begin{rem}
Let $(\ve_i)_{i\geq 1}$ be i.i.d.\ symmetric $\pm 1$-valued r.v.s, $X_t=\sum_{i=1}^\infty t_i\ve_i$, $t\in \ell^2$ and
$T=\{e_n\colon\ n\geq 1\}$, where $(e_n)$ is the canonical basis of $\ell^2$.
Then obviously $\Ex\sup_{s,t\in T}(X_s-X_t)=2$, moreover for any $A\subset T$
with cardinality at least 2, we have $\Delta_{2^k}(T)\geq \Delta_2(T)=\sqrt{2}$, hence $\gamma_X(T)=\infty$. Therefore
one cannot reverse bound \eqref{eq:upper} for Bernoulli processes, so some assumptions on the nontrivial speed of
growth of moments are necessary in Theorem \ref{thm:supcan}. However, Corollary \ref{cor:convhull} holds for Bernoulli
processes and we believe that in that statement the assumption of the $\beta$-speed of the moments growth is not needed.
\end{rem}

The crucial step in deriving lower bounds for suprema of stochastic processes is the Sudakov-minoration principle.
Following \cite{La_sud} (see also \cite{MMP}) we say that a process $(X_t)_{t\in S}$
satisfies \emph{the Sudakov minoration principle with constant} $\kappa>0$ if for any $p\geq 1$,
$T\subset S$ with $|T|\geq e^p$ such that $\|X_s-X_t\|_p\geq u$ for all $s,t\in T$, $s\neq t$,
we have $\Ex\sup_{s,t\in T}(X_s-X_t)\geq \kappa u$.

\begin{thm}
\label{thm:sud}
Suppose that $X_1,X_2,\ldots$ are independent standardized r.v.s  and moments of $X_i$ grow $\alpha$-regularly
for some $\alpha\geq 1$. Then the canonical process
$X_t=\sum_{i=1}^{\infty}t_iX_i$, $t\in \ell^2$ satisfies
the Sudakov minoration principle with  constant $\kappa(\alpha)$, which depends only on $\alpha$.
\end{thm}

In fact the assumption on regular growth of moments is necessary for the Sudakov minoration principle
in the i.i.d.\ case.

\begin{prop}
\label{prop:sud_yield_reg}
Suppose that a canonical process  $X_t=\sum_{i=1}^{\infty}t_iX_i$, $t\in  \ell^2$ based on i.i.d.\  standardized
random variables $X_i$  satisfies the Sudakov minoration with  constant $\kappa>0$.
Then  moments of $X_i$ grow $C/\kappa$-regularly.
\end{prop}

The next simple observation shows that (under mild regularity assumptions) the Sudakov minoration is necessary
for reversing bound \eqref{eq:upper}.

\begin{rem}
Suppose that for any finite $T\subset \ell^2$ we have $\Ex\sup_{s,t\in T}(X_s-X_t)\geq \kappa\gamma_X(T)$.
Assume moreover that for any $p\geq 1$ and $t\in \ell^2$, $\|X_t\|_{2p}\leq \gamma\|X_t\|_p$.
Then $X$ satisfies the Sudakov minoration principle with constant $\kappa/\gamma$.
\end{rem}

\begin{proof} Let $p\geq 1$ and $T\subset \ell^2$ of cardinality at least $e^p$ be such that
$\|X_s-X_t\|_p\geq u$ for any $s,t\in T$, $s\neq t$.
Let $2^k\leq p< 2^{k+1}$ and $({\cal A}_n)$ be an admissible sequence of partitions of the set $T$.
Then there is $A\in {\cal A}_k$ which contains at least two points of $T$.
Hence
\[
\Ex\sup_{s,t\in T}(X_s-X_t)\geq \kappa\gamma_X(T)\geq \kappa\Delta_{2^k}(A)\geq \kappa\Delta_{\max\{p/2,1\}}(A)\geq
\kappa u/\gamma.
\]
\end{proof}

In fact in the i.i.d.\ case we do not need the regularity assumption $\|X_t\|_{2p}\leq \gamma\|X_t\|_p$.

\begin{prop}
\label{prop:suptosud}
Let $X_t=\sum_{i=1}^{\infty}t_iX_i$, $t\in \ell^2$, where $X_i$ are i.i.d.\ standardized r.v.s.
Suppose that $\Ex\sup_{t,s\in T}X_t\geq \kappa\gamma_X(T)$ for all finite $T\subset \ell^2$. Then $(X_t)_{t\in \ell^2}$
satisfies  the Sudakov minoration principle with constant $\kappa/2$. In particular, moments of $X_i$
grow $C/\kappa$-regularly.
\end{prop}

Methods developed to prove Theorem \ref{thm:sud} also enable us to establish the following
comparison of weak and strong moments of the canonical processes based on variables with regular growth of moments.

\begin{thm}
\label{thm:weakstrong}
Let $X_t$ be as in Theorem \ref{thm:sud}. Then for any nonempty $T\subset \ell_2$ and $p\geq 1$,
\[
\left(\Ex\sup_{t\in T}|X_t|^p\right)^{1/p}\leq
C(\alpha)\left( \Ex\sup_{t\in T}|X_t|+\sup_{t\in T}(\Ex|X_t|^p)^{1/p}\right).
\]
\end{thm}

This paper is organized as follows. In the next section we gather some general facts. In Section \ref{sec:sudakov}
we study the class ${\mathcal R}_{\alpha}$ and show that variables in this class have tails comparable to
variables with log-concave tails. Based on this observation we establish the Sudakov minoration principle
(Theorem \ref{thm:sud}). We finish that section with
 the proofs of Theorem \ref{thm:weakstrong} and Proposition \ref{prop:sud_yield_reg}.
Section \ref{sec:lowerbounds} is devoted to reversing bound \eqref{eq:upper}. We obtain further regularity properties
of the tails of variables from class ${\mathcal R}_{\alpha} \cap {\mathcal S}_{\beta}$ and then prove
Theorem~\ref{thm:supcan} as well as
Corollaries \ref{cor:comptails} and \ref{cor:convhull}.
We close Section \ref{sec:lowerbounds} by  proving Proposition \ref{prop:suptosud}.

\subsection*{Notation} By $\ve_i$ we denote a Bernoulli sequence, i.e.\ a sequence of i.i.d.\ symmetric r.v.s taking values
$\pm 1$. We assume that variables $\ve_i$ are independent of other r.v.s. By a letter $C$ we denote universal
constants. Value of a constant $C$ may differ at each occurrence. Whenever we want to fix the value of an absolute
constant we use letters $C_1,C_2,\ldots$. We write $C(\alpha)$ (resp.\ $C(\alpha,\beta)$, etc.) for constants depending
only on parameters $\alpha$ (resp. $\alpha,\beta$ etc.).

\section{Preliminaries}\label{sec:prelim}

In this section we gather basic facts used in the sequel. We start with the contraction principle
for Bernoulli processes (see e.g. \cite[Theorem 4.4]{LT}).

\begin{thm}[Contraction principle]
\label{thm:contraction}
Let $(a_i)_{i=1}^n$, $(b_i)_{i=1}^n$ be two sequences of real numbers such that $|a_i| \leq |b_i|$, $i = 1, \ldots, n$.
Then
\begin{equation}
\label{eq:contraction1}
\Ex F\left(\left|\sum_{i=1}^n a_i\ve_i \right|\right) \leq \Ex F\left(\left|\sum_{i=1}^n b_i\ve_i \right|\right),
\end{equation}
where $F\colon\er_+\rightarrow\er_+$ is a convex function. In particular,
\begin{equation}
\label{eq:contraction2}
\left\|\sum_{i=1}^n a_i\ve_i\right\|_p \leq \left\|\sum_{i=1}^n b_i\ve_i\right\|_p.
\end{equation}
Moreover, for a nonempty subset $T$ of $\er^n$,
\begin{equation}
\label{eq:contraction3}
\Ex \sup_{t \in T} \sum_{i=1}^n t_ia_i\ve_i \leq \Ex \sup_{t \in T} \sum_{i=1}^n t_ib_i\ve_i.
\end{equation}
\end{thm}

The next Lemma is a standard symmetrization argument (see e.g. \cite[Lemma 6.3]{LT}).

\begin{lem}[Symmetrization]
\label{lem:symm}
Let $X_i$ be independent standardized r.v.s and $(\ve_i)$ be a Bernoulli sequence independent
of $(X_i)$. Define two canonical processes $X_t=\sum_{i=1}^\infty t_iX_i$ and its symmetrized version
$\tilde{X}_t=\sum_{i=1}^\infty t_i\ve_iX_i$. Then
\[
\frac{1}{2}\|X_s-X_t\|_p\leq \|\tilde{X}_s-\tilde{X}_t\|_p\leq 2\|X_s-X_t\|_p \quad \mbox{ for }s,t\in \ell^2
\]
and for any $T\subset \ell^2$,
\[
\frac{1}{2}\Ex\sup_{s,t\in T}(X_s-X_t)\leq \Ex\sup_{s,t\in T}(\tilde{X}_s-\tilde{X}_t)=
2\Ex\sup_{t\in T}\tilde{X}_t\leq 2\Ex\sup_{s,t\in T}(X_s-X_t).
\]
\end{lem}

Let us also recall the Paley-Zygmund inequality (cf. \cite[Lemma 0.2.1]{KW})
which goes back to work \cite{PZ} on trigonometric series.

\begin{lem}[Paley-Zygmund inequality]
For any nonnegative random variable $S$ and $\lambda\in (0,1)$,
\begin{equation}
\label{eq:PZ}
\Pr(S\geq \lambda \Ex S)\geq (1-\lambda)^2\frac{(\Ex S)^2}{\Ex S^2}.
\end{equation}
\end{lem}

The next lemma shows that convolution preserves (up to a universal constant) the property of the $\alpha$-regular
growth of moments.

\begin{lem}
\label{lem:convreg}
Let $S=\sum_{i=1}^n X_i$, where $X_i$ are independent mean zero r.v.s with moments growing $\alpha$-regularly. Then
moments of $S$ grow $C\alpha$-regularly. In particular, if $(X_t)$ is a  canonical process based on
r.v.s from ${\cal R}_{\alpha}$, then $\|X_t\|_{4p}\leq C\alpha \|X_t\|_p$ for $p\geq 2$.
\end{lem}

\begin{proof}
We are to show that $\|S\|_p\leq C\alpha\frac{p}{q}\|S\|_q$ for $p\geq q\geq 2$.
By Lemma \ref{lem:symm} we may assume that the r.v.s $X_i$ are symmetric. Moreover, by monotonicity of moments,
it is enough to consider only the case when $p$ and $q$ are even integers and $p\geq 2q$. In \cite{La_mom} it was shown
that for $r\geq 2$,
\[
\frac{e-1}{2e^2}|||(X_i)|||_r\leq \|S\|_r\leq e|||(X_i)|||_r,
\]
where
\[
|||(X_i)|||_r:=\inf\left\{u>0\colon\ \prod_{i}\Ex\left|1+\frac{X_i}{u}\right|^r\leq e^r\right\}.
\]
Therefore it is enough to proof the following claim.

\medskip

\noindent
{\bf Claim.} Suppose that $Y$ is a symmetric r.v.\ with moments growing $\alpha$-regularly. Let $p,q$ be positive even
integers such that $p\geq 2q$ and $\Ex|1+Y|^q\leq e^A\leq e^q$. Then $\Ex|1+\frac{q}{4e\alpha p}Y|^p\leq e^{pA/q}$.
\medskip

To show the claim first notice that
\[
\Ex|1+Y|^q=1+\sum_{k=1}^{q/2}\binom{q}{2k}\Ex|Y|^{2k}\geq 1+\sum_{k=1}^{q/2}\left(\frac{q}{2k}\right)^{2k}\Ex|Y|^{2k}
\geq 1+\Ex|Y|^q.
\]
In particular, $\|Y\|_q\leq (e^{A}-1)^{1/q}\leq e$.
On the other hand,
\[
\Ex\left|1+\frac{q}{4e\alpha p}Y\right|^p=1+\sum_{k=1}^{p/2}\binom{p}{2k}\Ex\left|\frac{q}{4e\alpha p}Y\right|^{2k}\leq
1+\sum_{k=1}^{p/2}\left(\frac{q}{8\alpha k}\right)^{2k}\Ex|Y|^{2k}.
\]
Since $\alpha\geq 1$ we obviously have
\[
1+\sum_{k=1}^{q/2}\left(\frac{q}{8\alpha k}\right)^{2k}\Ex|Y|^{2k}\leq \Ex|1+Y|^q\leq e^A.
\]
The $\alpha$-regularity of moments of $Y$ yields
\[
\sum_{k=q/2+1}^{p/2}\left(\frac{q}{8\alpha k}\right)^{2k}\Ex|Y|^{2k}\leq
\sum_{k=q/2+1}^{p/2} \left(\frac{1}{4}\|Y\|_q\right)^{2k}\leq
\left(\frac{1}{4}\|Y\|_q\right)^{q}\sum_{l=1}^{\infty}\left(\frac{e}{4}\right)^{2l}\leq \|Y\|_q^q.
\]
Thus
\[
\Ex\left|1+\frac{q}{4e\alpha p}Y\right|^p\leq e^A+\|Y\|_q^q\leq 2e^{A}-1\leq e^{2A}\leq e^{pA/q},
\]
which completes the proof of the claim and of the lemma.
\end{proof}

We finish this section with the observation that will allow us to compare regular r.v.s with variables
with log-concave tails.

\begin{lem}
\label{lm:convex}
Let a nondecreasing function $f\colon\er_+\rightarrow\er_+$ satisfy
\[
f(c\lambda t) \geq \lambda f(t), \qquad \mbox{for } \lambda \geq 1,\ t \geq t_0,
\]
where $t_0 \geq 0$, $c \geq 2$ are some constants. Then there is a function
$g\colon\er_+\rightarrow\er_+$, convex on $[ct_0, \infty)$, such that
\[
g(t) \leq f(t) \leq g(c^2t), \qquad \mbox{for }t \geq ct_0,
\]
and $g(ct_0) = 0$.
\end{lem}

\begin{proof}
For $t \geq ct_0$ we set
\[
g(t) := \int_{ct_0}^t \sup_{ct_0 \leq y \leq x} \frac{f(y/c)}{y} \dd x.
\]
Then $g$ is convex on $[ct_0, \infty)$ as an integral of a nondecreasing function. For $t \geq x\geq ct_0$ we have
$\sup_{ct_0 \leq y \leq x} f(y/c)/y \leq f(t)/t$, as $f(\lambda y)/(\lambda y) \geq f(y/c)/y$ for $y \geq ct_0$ and
$\lambda\geq 1$. Thus
\[
g(t) \leq (t-ct_0)\frac{f(t)}{t} \leq f(t), \qquad \mbox{for } t \geq ct_0.
\]
Moreover, for $t \geq ct_0$
\begin{align*}
g(ct)
&= \int_{ct_0}^{ct} \sup_{ct_0 \leq y \leq x} \frac{f(y/c)}{y} \dd x
\geq \int_t^{ct} \sup_{ct_0 \leq y \leq x} \frac{f(y/c)}{y} \dd x
\\
&\geq (ct-t)\frac{f(t/c)}{t} = (c-1)f(t/c) \geq f(t/c),
\end{align*}
hence $g(c^2t) \geq f(t)$ for $t \geq ct_0$.
\end{proof}

\section{Sudakov minoration principle}\label{sec:sudakov}

The main goal of this section is to prove Theorem \ref{thm:sud}. The strategy of the proof is to reduce the problem involving
random variables with moments growing regularly to the case of random variables with log-concave tails, for which the
minoration is known
(see \cite[Theorem 1]{La_sup}). The relevant result can be restated as follows

\begin{thm}
\label{thm:latala}
Let $X_t=\sum_{i=1}^\infty t_iX_i$, $t\in \ell^2$ be the canonical process based on independent symmetric r.v.s $X_i$
with log-concave tails. Then $(X_t)_{t\in \ell^2}$ satisfies the Sudakov minoration principle with
a universal constant $\kappa_{\mathrm{lct}}>0$.
\end{thm}

\begin{rem}
Since we may normalize $X_i$ we  do not need to assume that they have variance one. It suffices to have
$\sup_{i}\Var(X_i)<\infty$ in order that $X_t$ is well defined for $t\in \ell^2$.
\end{rem}

The mentioned reduction hinges on the idea that the tail functions of random variables with regular growth of moments
ought to be close to log-concave functions as, conversely, log-concave random variables are regular.

\begin{prop}
\label{prop:betweenconvex}
Let $\alpha \geq 1$. There exist constants $T_\alpha, L_\alpha$ such that for any $X \in \mathcal{R}_\alpha$
there is a nondecreasing function $M\colon [0,\infty)\rightarrow [0,\infty]$ which is convex, $M(T_\alpha) = 0$,
and satisfies
\begin{equation}
\label{eq:betweenconvex}
M(t) \leq N(t) \leq M(L_\alpha t), \qquad \mbox{for }t \geq T_\alpha,
\end{equation}
where $N(t) = -\ln\Pr(|X| > t)$.
\end{prop}

\begin{proof}
Fix $\alpha \geq 1$. We begin with showing that there is a constant $\kappa_\alpha$ such that for any
$X \in \mathcal{R}_\alpha$,
\begin{equation}
\label{eq:sublinear}
N(\kappa_\alpha \lambda t) \geq \lambda N(t), \qquad \lambda \geq 1,\ t \geq 1 - 1/e.
\end{equation}
When $\|X\|_\infty < \infty$ it is enough to prove this assertion for $t < (1-1/e)\|X\|_\infty$ as, providing that
$\kappa_\alpha \geq (1-1/e)^{-1}$, for $t \geq (1-1/e)\|X\|_\infty$ we have
$N(\kappa_\alpha \lambda t) \geq N\left( \n{X}_\infty \right) = \infty$.

So, fix $\lambda \geq 1$ and $1-1/e\leq t < (1-1/e)\|X\|_\infty$. There exists $q \geq 2$ such that $t = (1-1/e)\|X\|_q$.
Pick also $p \geq q$ so that $\lambda = p/q$. By the Paley-Zygmund inequality \eqref{eq:PZ} and
by the assumption that $X \in \mathcal{R}_\alpha$ we obtain
\begin{align}
\notag
N(t)
&= N\left( (1-1/e)\|X\|_q \right) \leq N\left( (1-1/e)^{1/q}\|X\|_q \right)
\\
\notag
&= -\ln \Pr(|X|^q > (1-1/e)\Ex|X|^q) \leq -\ln \left( \frac{1}{e^2}\left( \frac{\|X\|_q}{\|X\|_{2q}} \right)^{2q} \right)
\\
\label{eq:bc1}
&\leq 2 + q\ln\left[(2\alpha)^2\right] \leq q\ln\left(e(2\alpha)^2\right) =: qb_\alpha.
\end{align}
On the other hand, setting $\kappa_\alpha = e^{b_\alpha}(1-1/e)^{-1}\alpha$, with the aid of the assumption that
$X \in \mathcal{R}_\alpha$ and Chebyshev's inequality, we get
\begin{align}
\notag
N(\kappa_\alpha \lambda t)
&= N\left( e^{b_\alpha} \alpha\frac{p}{q}\|X\|_q \right) \geq N\left( e^{b_\alpha} \|X\|_p \right)
\\
\label{eq:bc2}
&= -\ln \Pr(|X|^p > e^{pb_\alpha} \Ex|X|^p) \geq pb_\alpha = \lambda qb_\alpha.
\end{align}
Joining inequalities \eqref{eq:bc1} and \eqref{eq:bc2} we get \eqref{eq:sublinear} with $\kappa_\alpha = \frac{4e^2}{e-1}\alpha^3$.

By virtue of this \emph{sublinear} property \eqref{eq:sublinear}, Lemma \ref{lm:convex} applied to $f = N$,
$c = \kappa_\alpha$, and $t_0 = 1-1/e$ finishes the proof, providing the constants
\[
L_\alpha = \kappa_\alpha^2 = \left(\frac{4e^2}{e-1}\alpha^3\right)^2, \qquad T_\alpha = \kappa_\alpha t_0 = 4e\alpha^3.
\]
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:sud}]
We fix $p\geq 2$, $T\subset \ell^2$ such that $|T|\geq e^p$ and $\|X_s-X_t\|_p\geq u$ for all distinct $s,t\in T$.
We are to show that $\Ex\sup_{s,t\in T}(X_s-X_t)\geq \kappa_{\alpha}u$
for a constant $\kappa_{\alpha}$ which depends only on $\alpha$.
By Lemma \ref{lem:symm} we may assume that r.v.s $X_i$ are symmetric.

Proposition \ref{prop:betweenconvex} yields that the tail functions $N_i(t) := -\ln \Pr(|X_i| > t)$
of the variables $X_i$ are controlled by the convex functions $M_i(t)$, apart from $t \leq T_\alpha$, i.e. we have
$M_i(t) \leq N_i(t) \leq M_i(L_\alpha t)$ only for $t\geq T_{\alpha}$.
To gain control also for $t \leq T_{\alpha}$, define the symmetric random variables
\[
\wt{X}_i = (\sgn X_i)\max\{|X_i|,T_\alpha\},
\]
so that their tail functions $\wt{N}_i(t) = -\ln\Pr(|\wt{X}_i|>t)$,
\[
\wt{N_i}(t)=
\begin{cases}
0, & t < T_{\alpha}
\\
N_i(t), & t \geq T_{\alpha}
\end{cases},
\]
satisfy
\begin{equation}
\label{eq:compofNandM}
M_i(t) \leq \wt{N}_i(t) \leq M_i(L_\alpha t) \qquad \mbox{for all }t \geq 0.
\end{equation}
This allows us to construct a sequence $Y_1, Y_2, \ldots$ of independent symmetric r.v.s with log-concave tails
given by $\Pr(|Y_i| > t) = e^{-M_i(t)}$ such that
\begin{equation}
\label{eq:YXtildeequiv}
|Y_i| \geq |\wt{X}_i| \geq \frac{1}{L_\alpha}|Y_i|.
\end{equation}
Define the canonical processes $\wt{X}_t:=\sum_{i=1}^{\infty}t_i\wt{X}_i$ and $Y_t:=\sum_{i=1}^{\infty}t_iY_i$, $t\in \ell^2$.

Since $|Y_i|\geq |X_i|$ and variables $Y_i$ and $X_i$ are symmetric we get for $s,t\in T$, $s\neq t$,
\[
\|Y_s-Y_t\|_p= \left\|\sum_{i=1}^\infty (s_i-t_i)|Y_i|\ve_i\right\|_p
\geq \left\|\sum_{i=1}^\infty (s_i-t_i)|X_i|\ve_i\right\|_p=\|X_s-X_t\|_p\geq u,
\]
where the first inequality follows by contraction principle \eqref{eq:contraction2} as $|Y_i| \geq |\wt{X}_i| \geq |X_i|$.
Hence we can apply Theorem \ref{thm:latala} to the canonical process $(Y_t)$ and
obtain
\begin{equation}
\label{eq:sudY}
2\Ex\sup_{t\in T}Y_t=\Ex \sup_{s,t \in T}(Y_s-Y_t) \geq \kappa_{\mathrm{lct}}u.
\end{equation}

To finish the proof it suffices to show that $\E\sup_{t \in T} X_t$ majorizes $\E\sup_{t\in T} Y_t$.
Clearly,
\begin{equation}\label{eq:obvtriangle}
\Ex\sup_{t \in T} X_t \geq \Ex\sup_{t \in T} \wt{X}_t - \Ex\sup_{t \in T} (\wt{X}_t-X_t).
\end{equation}
Recall that by the definition of $\wt{X}_i$,
$|\wt{X}_i - X_i| = |T_\alpha - X_i|\1_{\{|X_i|\leq T_\alpha\}} \leq T_\alpha$.
As a consequence, the supremum of the canonical process $\E\sup_{t \in T} (\wt{X}_t-X_t)$ is bounded by the supremum of
the Bernoulli process $\E\sup_{t \in T} \sum t_iT_{\alpha}\ve_i$. Indeed, using the symmetry of the distribution of
the variables $\wt{X}_i - X_i$ and contraction principle \eqref{eq:contraction3},
\[
\Ex\sup_{t \in T} (\wt{X}_t-X_t) = \Ex_X\Ex_\ve\sup_{t \in T} \sum_{i=1}^{\infty} t_i|\wt{X}_i-X_i|\ve_i
\leq \Ex_\ve \sup_{t \in T} \sum_{i=1}^{\infty} t_iT_\alpha\ve_i.
\]

Since $X_i\in {\cal R}_{\alpha}$ we get by H\"older's inequality,
\begin{align*}
1 = \Ex X_i^2 = \Ex X_i^{4/3}X_i^{2/3} \leq \|X_i\|_4^{4/3}\|X_i\|_1^{2/3} \leq (2\alpha\|X_i\|_2)^{4/3}\|X_i\|_1^{2/3}=(2\alpha)^{4/3}(\Ex|X_i|)^{2/3}
\end{align*}

and thus $\Ex|X_i|\geq (2\alpha)^{-2}$. Hence by Jensen's inequality
\[
\Ex\sup_{t \in T} X_t = \Ex_\ve\Ex_X \sup_{t\in T} \sum_{i=1}^{\infty} t_i|X_i|\ve_i
\geq \Ex_\ve \sup_{t \in T} \sum_{i=1}^{\infty} t_i\Ex_X|X_i|\ve_i \geq
\frac{1}{(2\alpha)^2} \Ex \sup_{t \in T} \sum_{i=1}^{\infty} t_i\ve_i.
\]
As a result,
\[
\Ex\sup_{t \in T} (\wt{X}_t-X_t) \leq (2\alpha)^2 T_\alpha \E\sup_{t \in T} X_t,
\]
and by \eqref{eq:obvtriangle},
\begin{equation}
\label{eq:supXtildeX}
\Ex\sup_{t \in T} X_t \geq \frac{1}{1+(2\alpha)^2 T_\alpha} \Ex\sup_{t \in T} \wt{X}_t.
\end{equation}
Finally, notice that, by virtue of contraction principle \eqref{eq:contraction3}, the second inequality of
\eqref{eq:YXtildeequiv} implies that
\begin{equation}
\label{eq:suptileXY}
\Ex\sup_{t \in T} \wt{X}_t \geq \frac{1}{L_\alpha} \Ex\sup_{t \in T} Y_t.
\end{equation}

Estimates \eqref{eq:sudY}, \eqref{eq:supXtildeX} and \eqref{eq:suptileXY} yield
\[
\Ex\sup_{s,t \in T}(X_s-X_t)=2\Ex\sup_{t \in T} X_t\geq \frac{2}{L_{\alpha}(1+(2\alpha)^2 T_\alpha)}\Ex\sup_{t \in T} Y_t
\geq \frac{\kappa_{\mathrm{lct}}}{L_{\alpha}(1+(2\alpha)^2 T_\alpha)}u.
\]
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:weakstrong}]
Using a symmetrization argument we may assume that the variables $X_i$ are symmetric. Let variables
$\wt{X}_i, Y_i$ and the related canonical processes be as in the proof of Theorem \ref{thm:sud}.
Since the variables $Y_i$ have log-concave tails by \cite{La_SM} we get
\[
\left(\Ex\sup_{t\in T}|Y_t|^p\right)^{1/p}\leq
C\left( \Ex\sup_{t\in T}|Y_t|+\sup_{t\in T}(\Ex|Y_t|^p)^{1/p}\right).
\]
Estimate $|Y_i|\geq |X_i|$ and the contraction principle yield
\[
\Ex\sup_{t\in T}|X_t|^p\leq \Ex\sup_{t\in T}|Y_t|^p.
\]
We showed above that
\[
\Ex\sup_{t\in T}|Y_t|\leq L_\alpha(1+(2\alpha)^2T_\alpha)\Ex\sup_{t\in T}|X_t|.
\]
Finally the contration principle together with the bounds $|Y_i|\leq L_{\alpha}|\wt{X}_i|$, $|X_i-\wt{X}_i|\leq T_{\alpha}$
and $\Ex |X_i|\geq (2\alpha)^{-2}$ imply
\[
\|Y_t\|_p\leq L_{\alpha}\|\wt{X}_t\|_p\leq L_{\alpha}\|X_t\|_p+L_{\alpha}T_{\alpha}\left\|\sum_{i=1}^{\infty}t_i\ve_i\right\|_p\leq
L_\alpha(1+T_{\alpha}(2\alpha)^2)\|X_t\|_p.
\]
\end{proof}

We conclude this section  with the proof of Proposition \ref{prop:sud_yield_reg} showing that in the i.i.d.\ case
the Sudakov minoration principle and the $\alpha$-regular growth of moments are  equivalent.

\begin{proof}[Proof of Proposition \ref{prop:sud_yield_reg}]
Let us fix $p\geq q\geq 2$ and for $1\leq m\leq n$ consider the following subset of $\ell^2$
\[
T=T(m,n) = \left\{t \in \ell^2\colon\ \sum_{i=1}^n t_i = m, \ t_i = 0, i > n\right\}.
\]
Then $|T|=\binom{n}{m}\geq (n/m)^m\geq e^p$ if $n\geq me^{p/m}$.
Moreover, for any $s, t \in T$, $s \neq t$, say with $s_j \neq t_j$ we have $\|X_s-X_t\|_p \geq \|X_j\|_p$.
Thus the Sudakov minoration principle yields for any $n\geq me^{p/m}$,
\begin{equation}
\label{eq:sud_yields_reg1}
\kappa\|X_i\|_p
\leq \Ex\sup_{s,t \in T} (X_s-X_t) \leq 2\Ex\sup_{\substack{I \subset [n] \\ |I| = m}} \sum_{i \in I} |X_i|
=2\Ex\sum_{k=1}^m X_k^*,
\end{equation}
where $(X_1^*,X_2^*,\ldots,X_n^*)$ is the nonincreasing rearrangement of $(|X_1|,|X_2|,\ldots,|X_n|)$.

We have
\begin{align*}
\Pr(X_k^*\geq t)=\Pr\left(\sum_{i=1}^n\1_{\{|X_i|\geq t\}}\geq k\right)\leq \frac{1}{k}\sum_{i=1}^n\Ex \1_{\{|X_i|\geq t\}}
=\frac{n}{k}\Pr(|X_i|\geq t)\leq \frac{n}{k}\frac{\|X_i\|_q^q}{t^q}.
\end{align*}
Integration by parts shows that
\[
\Ex X_k^*\leq C\left(\frac{n}{k}\right)^{1/q}\|X_i\|_q.
\]
Combining this with \eqref{eq:sud_yields_reg1} we get (recall that $q\geq 2$ and constant $C$ may differ at each
occurrence)
\[
\kappa\|X_i\|_p\leq C\sum_{k=1}^m\left(\frac{n}{k}\right)^{1/q}\|X_i\|_q\leq Cn^{1/q}m^{1-1/q}\|X_i\|_q.
\]

Taking $m=\lceil p/q\rceil$ and $n=\lceil me^{p/m}\rceil$ we find that
$n^{1/q}m^{1-1/q} \leq 4ep/q$. Hence
\[
\|X_i\|_p \leq \frac{C}{\kappa}\frac{p}{q}\|X_i\|_q
\]
which finishes the proof.
\end{proof}

\section{Lower bounds for suprema of canonical processes}
\label{sec:lowerbounds}

As in the case of the Sudakov minoration principle the proof of the lower bound in Theorem \ref{thm:supcan} is based
on the corresponding result for the canonical processes built on variables with log-concave tails.
Theorem 3 in \cite{La_sup} (see also Theorem 10.2.7 and Exercise 10.2.14  in \cite{Talnew}) implies the following result.

\begin{thm}
\label{thm:suplct}
Let $X_t=\sum_{i=1}^\infty t_iX_i$, $t\in \ell^2$ be the canonical process based on independent symmetric r.v.s $X_i$
with log-concave tails. Assume moreover that there exists $\gamma$ such that $N_i(2t)\leq \gamma N_i(t)$ for all
$i$ and $t>0$, where $N_i(t)=-\ln\Pr(|X_i|> t)$. Then there exists a constant $C_{\mathrm{lct}}(\gamma)$, which
depends only on $\gamma$ such that for any $T\subset \ell^2$,
\[
\Ex\sup_{s,t\in T}(X_s-X_t)=2\Ex\sup_{t\in T}X_t\geq \frac{1}{C_{\mathrm{lct}}(\gamma)}\gamma_X(T).
\]
\end{thm}

\begin{rem}
Theorem 3 in \cite{La_sup} and Theorem 10.2.7 in \cite{Talnew}  were formulated in a slightly different language.
In particular, the latter states that there exist $r>2$, an admissible sequence of partitions $({\cal A}_{n})$
and numbers $j_n(A)$ for $A\in {\cal A}_n$ such that $\varphi_{j_n(A)}(s,s')\leq 2^{n+1}$ for all
$s,s'\in A$  and
\[
\sup_{t\in T}\sum_{n=0}^{\infty}2^{n}r^{-j_n(A_n(t))}\leq C(\gamma)\Ex\sup_{t\in T} X_t.
\]
 (For the definition of $\varphi$ see \cite{Talnew} - it precedes the statement of Theorem 10.2.7.) However, the condition $\varphi_{j_n(A)}(s,s')\leq 2^{n+1}$ yields that $\|X_s-X_{s'}\|_{2^{n}}\leq C2^nr^{-j_n(A)}$
(see \cite{GK} for the i.i.d. case and Example 3 in \cite{La_mom} for the general situation),
so $\Delta_{2^{n}}(A_n(t))\leq C2^nr^{-j_n(A_n(t))}$ and
\[
\gamma_X(T)\leq C\sup_{t\in T}\sum_{n=0}^{\infty}2^{n}r^{-j_n(A_n(t))}\leq C_{\mathrm{lct}}(\gamma)\Ex\sup_{t\in T} X_t.
\]
\end{rem}

\begin{prop}
\label{prop:taillowgrowth}
Let $\alpha\geq 1, \beta > 1$. For any $r > 1$ there exists a constant $C(\alpha,\beta,r)$ such that
for  $X \in \mathcal{R}_\alpha \cap \mathcal{S}_\beta$ we have
\begin{equation}
N(rt) \leq C(\alpha,\beta,r) N(t), \qquad t \geq 2,
\end{equation}
where $N(t) := -\ln \Pr(|X| > t)$.
\end{prop}

\begin{proof}
Fix $t \geq 2$ and define
\[
q:=\inf\{p\geq 2\colon\ \|X\|_{\beta p}\geq t\}.
\]
Since $X \in \mathcal{R}_\alpha \cap \mathcal{S}_\beta$, the function $p\longmapsto \|X\|_p$ is finite
and continuous on $[2,\infty)$, moreover $\|X\|_2=1$ and $\|X\|_{\infty}=\infty$. Hence, if
$t\geq \|X\|_{2\beta}$, we have $t=\|X\|_{\beta q}$ and  by Chebyshev's inequality,
\[
N(t) = N( \|X\|_{\beta q}) \geq N(2 \|X\|_q) = -\ln \Pr(|X|^q > 2^q\E|X|^q)
\geq q\ln 2.
\]
If $2\leq t<\|X\|_{2\beta}$, then $q=2$ and
\[
N(t)\geq N(2)= -\ln \Pr(|X|^2 > 4\Ex|X|^2)  \geq \ln 4=q\ln 2.
\]

Set an integer $k$ such that $r \leq 2^{k-2}$. Then, using consecutively the definition of $q$, the assumption that
$X \in \mathcal{S}_\beta$, the Paley-Zygmund inequality, and the assumption that $X \in \mathcal{R}_\alpha$,
we get the estimates
\begin{align}
\notag
N(rt)
&\leq N\left( 2^{k-2}\|X\|_{\beta q} \right) \leq N\left(\frac{1}{2}\|X\|_{\beta^kq} \right)
= -\ln \Pr\left( |X|^{\beta^kq} > 2^{-\beta^k q}\Ex|X|^{\beta^kq} \right)
\\
\label{eq:tailgr}
&\leq -\ln \left( \frac{1}{4} \left( \frac{\|X\|_{\beta^kq}}{\|X\|_{2\beta^{k}q}} \right)^{2\beta^kq} \right)
\leq \ln 4 + 2\beta^kq\ln (2\alpha)\leq q(\ln 2+2\beta^k\ln(2\alpha)).
\end{align}

Combining the above estimates we obtain the assertion with
$C(\alpha,\beta,r) =(\ln 2+2\beta^k\ln(2\alpha))/\ln 2$ and
 $k = k(r)$ being an integer such that $2^{k-2} \geq r$.
\end{proof}

\begin{remark}
\label{rem:Nbound}
Taking in \eqref{eq:tailgr} $t = 2$ which corresponds to $q = 2$ we find that
\[
N(s) \leq 2(\ln 2+2\beta^k\ln(2\alpha)),\qquad \mbox{for }s<2^{k-1},
\]
which means that the tail distribution function of a variable $X \in \mathcal{R}_\alpha\cap \mathcal{S}_\beta$
at a certain value $s$ is bounded with a constant not depending on the distribution of $X$ but only on the
parameters $\alpha, \beta$ and of course the value of $s$.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm:supcan}]
In view of \eqref{eq:upper} we are to address only the lower bound on $\Ex \sup_{t \in T} X_t$. A symmetrization argument
shows that we may assume that variables $X_i$ are symmetric.

Given symmetric $X_i$ let $Y_i$ be random variables defined as in the proof  of Theorem \ref{thm:sud},
i.e.\ $Y_i$'s are independent symmetric r.v.s having log-concave tails $\Pr(|Y_i| > t) = e^{-M_i(t)}$.
Moreover, let $L_\alpha, T_\alpha$ be the constants as in Proposition \ref{prop:betweenconvex}.
Due to Proposition \ref{prop:taillowgrowth} for $r = 2L_\alpha$ we know that the functions $N_i(t) := -\Pr(|X_i| > t)$
satisfy
\[
N_i(2L_\alpha t) \leq \gamma N(t), \qquad t \geq 2,
\]
where $\gamma=\gamma(\alpha,\beta):=C(\alpha,\beta,2L_{\alpha})$.

What then can be said about $M_i$? Using \eqref{eq:betweenconvex} we find that for
$t \geq \tilde{T}_\alpha:=\max\{2,T_{\alpha}\}$
\[
M_i(2L_\alpha t) \leq N_i(2L_\alpha t) \leq \gamma N_i(t) \leq \gamma M_i(L_\alpha t),
\]
which means that $M_i$ are almost of moderate growth, namely for $t_{\alpha}:=L_\alpha\tilde{T}_\alpha$ we have
\[
M_i(2t) \leq \gamma M_i(t), \qquad t \geq t_\alpha.
\]
Therefore, we improve the function $M_i$ putting on the interval $[0, t_\alpha]$ an artificial linear piece
$t \mapsto \lambda(i,\alpha) t$, where $\lambda(i,\alpha):= M_i(t_\alpha)/t_\alpha$. In other words, take the numbers
$p(i,\alpha) := \Pr(|Y_i| > t_\alpha) = e^{-M_i(t_\alpha)}$ and let $U_i$ be a sequence of independent random variables
with the following symmetric \emph{truncated} exponential distribution,
\[
\Pr(|U_i| > t) =
\begin{cases}
\frac{e^{-\lambda(i,\alpha) t}-p(i,\alpha)}{1-p(i,\alpha)}, & t \leq t_\alpha
\\
0, & t > t_\alpha
\end{cases},
\]
which are in addition independent of the sequences $(X_i)$ and $(Y_i)$. Define
\[
Z_i := Y_i\1_{\{|Y_i| > t_\alpha\}} + U_i\1_{\{|Y_i| \leq t_\alpha\}}.
\]
Let
\[
\wt{M}_i(t): = -\ln \Pr(|Z_i| > t) =
\begin{cases}
\lambda(i,\alpha) t, & t \leq t_\alpha,
\\
M_i(t), & t > t_\alpha.
\end{cases}
\]
Then $\wt{M}_i$ are convex functions of moderate growth, i.e.
\[
\wt{M}_i(2t) \leq \tilde{\gamma} \wt{M}_i(t), \qquad t \geq 0,
\]
where $\tilde{\gamma}=\tilde{\gamma}(\alpha,\beta):=\max\{2,\gamma\}$.

Thus Theorem \ref{thm:suplct} can be applied to the canonical process $Z_t:=\sum_{i}t_iZ_i$ and we get
\[
\Ex \sup_{t\in T}Z_t\geq \frac{1}{C_1(\alpha,\beta)}\gamma_Z(T),
\]
where $C_1(\alpha,\beta)=C_{\mathrm{lct}}(\tilde{\gamma})$.

What is left is to compare both the suprema and the functionals $\gamma$'s of the processes $(X_t)$ and $(Z_t)$.
The former is easy, because we have $M_i(t) \leq \wt{M}_i(t)$, $t \geq 0$, which allows to take samples such
that $|Y_i| \geq |Z_i|$, and consequently, thanks to contraction principle \eqref{eq:contraction3},
$\Ex\sup_{t \in T} Z_t \leq \Ex\sup_{t \in T} Y_t$. Joining this with estimates \eqref{eq:suptileXY} and
\eqref{eq:supXtildeX} we derive
\[
\Ex \sup_{t \in T} Z_t \leq L_\alpha(1 + (2\alpha)^2 T_\alpha) \E \sup_{t \in T} X_t.
\]

For the latter, we would like to show $C(\alpha,\beta)\gamma_Z \geq \gamma_X$ . It is enough to compare the metrics,
 i.e.\ to prove that $C(\alpha,\beta)\|Z_s-Z_t\|_p \geq \|X_s-X_t\|_p$ for $p\geq 1$. We proceed as in the proof of
 Theorem \ref{thm:sud}. We have
\begin{equation}
\label{eq:momentsZtriang}
\|Z_s-Z_t\|_p \geq \|Y_s-Y_t\|_p - \|(Y_s-Z_s)-(Y_t-Z_t)\|_p.
\end{equation}
In the proof of Theorem \ref{thm:sud} it was established that $\|Y_s-Y_t\|_p \geq \|X_s-X_t\|_p$.
For the second term we use the symmetry of the variables $Y_i-Z_i$, contraction principle \eqref{eq:contraction2},
and the fact that $|Y_i - Z_i| \leq 2t_\alpha$, obtaining
\begin{equation}
\label{eq:momentsY-Z}
 \|(Y_s-Z_s)-(Y_t-Z_t)\|_p = \left\|\sum_{i}(s_i-t_i)|Y_i - Z_i|\ve_i\right\|_p
 \leq 2t_\alpha\left\|\sum_i (s_i-t_i)\ve_i\right\|_p.
\end{equation}
Now we compare $\|Z_s-Z_t\|_p$ with moments of increments of the Bernoulli process. By Jensen's inequality we get
\begin{equation}
\label{eq:momentsZ}
\|Z_s-Z_t\|_p = \left\|\sum_i (s_i-t_i)|Z_i|\ve_i\right\|_p \geq
\min_i \Ex|Z_i| \left\|\sum_i (s_i-t_i)\ve_i\right\|_p.
\end{equation}
Combining \eqref{eq:momentsZtriang}, \eqref{eq:momentsY-Z}, and \eqref{eq:momentsZ} yields
\[
\|Z_s-Z_t\|_p \geq \left( 1 + \frac{2t_\alpha}{\min_i\Ex |Z_i|} \right)^{-1}\|X_s-X_t\|_p.
\]
To finish it suffices to prove that $\Ex|Z_i| \geq c_{\alpha,\beta}$ for some positive constant $c_{\alpha,\beta}$, which
depends only on $\alpha$ and $\beta$. This is a cumbersome yet simple calculation.
Recall the distributions of the variables $Y_i$ and $U_i$,
the fact that they are independent, and observe that
\begin{align*}
\Ex|Z_i|
&= \E|Y_i|\1_{\{|Y_i| > t_\alpha\}} + \Ex|U_i|\1_{\{|Y_i| \leq t_\alpha\}}
\\
&\geq t_\alpha \Pr(|Y_i| > t_\alpha) + \left( \Ex|U_i| \right)\Pr(|Y_i| \leq t_\alpha)
\\
&= t_\alpha p(i,\alpha) + (1 - p(i,\alpha))\int_0^{t_\alpha} \frac{e^{-\lambda(i,\alpha) t}-p(i,\alpha)}{1-p(i,\alpha)} \dd t
\\
&=\frac{1}{\lambda(i,\alpha)}\left( 1 - e^{-\lambda(i,\alpha) t_\alpha} \right) =
\frac{t_\alpha}{M_i(t_\alpha)}\left( 1 - e^{-M_i(t_\alpha)} \right).
\end{align*}
The last expression is nonincreasing with respect to $M_i(t_\alpha)$.
Since $M_i(t_\alpha) \leq N_i(t_\alpha)$ (see \eqref{eq:betweenconvex}), we are done provided that we can bound $N_i(t_\alpha)$ above.
Thus, Remark \ref{rem:Nbound} completes the proof.
\end{proof}

\begin{proof}[Proof of Corollary \ref{cor:comptails}]
Proposition 20 in \cite{La_sud} yields for $p\geq 1$,
\begin{align*}
\left(\Ex\sup_{t,s\in T}|Y_t-Y_s|^p\right)^{1/p}&\leq C(\gamma_Y(T)+\sup_{s,t\in T}\|Y_s-Y_t\|_p)\leq
C(\gamma_X(T)+\sup_{s,t\in T}\|X_s-X_t\|_p)
\\
& \leq C(\alpha,\beta) \left(\Ex\sup_{s,t\in T}|X_s-X_t|+\sup_{s,t\in T}\|X_s-X_t\|_p\right)
\\
&\leq
(C(\alpha,\beta)+1)\left\|\sup_{s,t\in T}|X_s-X_t|\right\|_{p},
\end{align*}
where the third inequality follows by Theorem \ref{thm:supcan}.
Hence by Chebyshev's inequality we obtain
\begin{equation}
\label{eq:estY}
\Pr\left(\sup_{t,s\in T}|Y_t-Y_s|\geq C_1(\alpha,\beta)\left\|\sup_{s,t\in T}|X_s-X_t|\right\|_{p}\right)
\leq e^{-p}
\quad \mbox{ for }p\geq 1.
\end{equation}

Theorem \ref{thm:weakstrong} (used for the set $T-T$) and Lemma \ref{lem:convreg} yield for $p\geq q\geq 1$,
\[
\left\|\sup_{s,t\in T}|X_s-X_t|\right\|_{p}\leq C_2(\alpha)\frac{p}{q}\left\|\sup_{s,t\in T}|X_s-X_t|\right\|_{q}.
\]
Hence, by the Paley-Zygmund inequality we get for $q\geq 1$,
\[
\Pr\left(\sup_{t,s\in T}|X_t-X_s|\geq \frac{1}{2}\left\|\sup_{s,t\in T}|X_s-X_t|\right\|_{q}\right)
\geq \frac{1}{4}\left(\frac{1}{2C_2(\alpha)}\right)^q.
\]
Applying the above estimate with $q=p/\ln(2C_2(\alpha))$ we get
\begin{equation}
\label{eq:estX}
\Pr\left(\sup_{t,s\in T}|X_t-X_s|\geq \frac{1}{2C_2(\alpha)\ln(2C_2\alpha)}
\left\|\sup_{s,t\in T}|X_s-X_t|\right\|_{p}\right)
\geq \frac{1}{4}e^{-p} \quad \mbox{ for }p\geq \ln(2C_2(\alpha)).
\end{equation}

The assertion easily follows by \eqref{eq:estY} and \eqref{eq:estX}.

\end{proof}

\begin{proof}[Proof of Corollary \ref{cor:convhull}]
By Theorem \ref{thm:supcan} we may find an admissible sequence of partitions $({\cal A}_n)$ such that
\begin{equation}
\label{eq:part}
\sup_{t\in T}\sum_{n=0}^\infty \Delta_{2^n}(A_n(t))\leq C(\alpha,\beta)\Ex\sup_{s,t\in T}(X_s-X_t).
\end{equation}
For any $A\in {\cal A}_n$ let us choose a point $\pi_n(A)\in A$ and set $\pi_n(t):=\pi_n(A_n(t))$.

Let $M_n:=\sum_{j=0}^{n} N_j$ for $n=0,1,\ldots$ (recall that we denote $N_j = 2^{2^j}$ for $j\geq 1$ and $N_0=1$).
Then $\log(M_n+2)\leq 2^{n+1}$.
Notice that there are $|{\cal A}_n|\leq N_n$ points of the form $\pi_n(t)-\pi_{n-1}(t)$, $t\in T$.
So we may set $s^1:=0$ and for $n=1,2,\ldots$ define
$s^{k}$, $M_{n-1}<k\leq M_n$ as some rearrangement (with repetition if $|{\cal A}_n|<N_n$)
of points of the form $(\pi_n(t)-\pi_{n-1}(t))/d_{2^{n+1}}(\pi_n(t),\pi_{n-1}(t))$, $t\in T$.
Then $\|X_{s^k}\|_{\log(k+2)}\leq 1$ for all $k$.

Observe that
\[
\|t-\pi_n(t)\|_2=\|X_t-X_{\pi_n(t)}\|_{2}\leq \Delta_2(A_n(t))\leq \Delta_{2^n}(A_n(t))\rightarrow 0
\mbox{ for }n\rightarrow\infty.
\]
For any $s,t\in T$ we have $\pi_0(s)=\pi_0(t)$ and thus
\[
s-t=\lim_{n\rightarrow\infty}(\pi_n(s)-\pi_n(t))=
\lim_{n\rightarrow\infty}\left( \sum_{k=1}^n(\pi_k(s)-\pi_{k-1}(s))-\sum_{k=1}^n(\pi_k(t)-\pi_{k-1}(t))\right).
\]
This shows that
\[
T-T\subset R\ \overline{\mathrm{conv}}\{\pm s^k\colon\ k\geq 1\},
\]
where
\begin{align*}
R&:=2\sup_{t\in T}\sum_{n=1}^{\infty}d_{2^{n+1}}(\pi_n(t),\pi_{n-1}(t))
\leq 2\sup_{t\in T}\sum_{n=1}^{\infty}\Delta_{2^{n+1}}(A_{n-1}(t))
\\
&\leq  C(\alpha)\sup_{t\in T}\sum_{n=1}^{\infty}\Delta_{2^{n-1}}(A_{n-1}(t))
\leq C(\alpha,\beta)\Ex\sup_{s,t\in T}(X_s-X_t),
\end{align*}
where the second inequality follows by Lemma \ref{lem:convreg} and the last one by \eqref{eq:part}.
Thus it is enough to define $t^k:=Rs^k$, $k\geq 1$.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:suptosud}]
Fix $p\geq 1$ and $T\subset \ell^2$ such that $|T|\geq e^p$ and $\|X_s-X_t\|_p\geq u$ for distinct points $s,t\in T$.
For $t^1,t^2\in T$ define a new point in $\ell^2$ by $t(t^1,t^2):=(t^1_1,t^2_1,t^1_2,t^2_2,\ldots)$.
Put also $\wt{T}:=\{t(t^1,t^2)\colon t^1,t^2\in T\}$. It is not hard to see that
$\|X_s-X_t\|_p\geq u$ for $t,s\in \wt{T}$, $t\neq s$.

Choose an integer $k$ such that $2^k\leq  p< 2^{k+1}$ and let $({\cal A}_n)$ be an admissible sequence of partitions
of the set $\wt{T}$. Since $|\wt{T}|=|T|^2\geq e^{2p}> 2^{2^{k+1}}$,
there is $A\in {\cal A}_k$ which contains at least two points of $\tilde{T}$.
Hence
\[
u\leq \Delta_{2^k}(A)\leq \gamma_X(\wt{T})\leq \frac{1}{\kappa} \Ex\sup_{s,t\in \wt{T}}(X_s-X_t)\leq
\frac{2}{\kappa} \Ex\sup_{s,t\in T}(X_s-X_t).
\]
\end{proof}

\section*{Acknowledgements}

The second named author would like to thank Filip Borowiec for a fruitful discussion regarding Lemma \ref{lm:convex}.

config: arxiv


TITLE: If I know in advance I'm getting an interference pattern, wouldn't I have time to remove the splitter so the which-path information would be know?
QUESTION [0 upvotes]: Check the image below, from this excellent explanation of the quantum delayed eraser. My question, and please forgive me if it is a naive one, is that first the photon reaches detector A, which tells us whether we have an interference pattern or not. With this information in hand, and because the second detection to know the which-path information is delayed for minutes or even millions of years, won't we have enough time to just remove the splitters and always send the entangled particle to detectors 2 and 3, effectively revealing the which-path information and causing a contradiction?
My point is that if I first look to find an interference pattern on detector A and then wait for a couple of millions years so I have time to effectively remove the splitters before going back to look again at detector A, will I find that my first observation has mysterious changed to the double line pattern, in other words, the future has just changed the past because what I first saw with my own eyes was an interference pattern that has now turned to a double line pattern?
Has anyone tried this experiment?

REPLY [1 votes]: The video doesn't give an explanation of the delayed choice quantum eraser - that is, an account of what is happening in reality to produce the results of the experiment. The video states the results of the experiment unclearly. The original paper
https://arxiv.org/abs/quant-ph/9903047
or even the wikipedia entry on this experiment:
https://en.wikipedia.org/wiki/Delayed-choice_quantum_eraser
are less misleading.
When the photon is measured at detector 1 all you see is a blob. If you only looked at the results from that detector you would never know what happened to the other photon. If you compare the results from detector 1 to those from detector 4 then you see that the probability of a photon arriving at the same position varies in an interference pattern with position. If the photon arrives at detectors 2 and 3 then you don't get that pattern in the comparison, but the pattern detected by measuring the results of detector 1 alone are just a blob. So if you remove the splitters and the photons arrive only at detectors 2 and 3 there will be no way to know that by looking at the results of measurements at detector 1 alone since the interference pattern or lack thereof is a pattern you get when you compare the results of measurements at detector 1 to detectors 2,3 and 4.
I think the explanation for these results is that all of the possible outcomes happen and the correlations are established when the measurement results are compared:
https://arxiv.org/abs/quant-ph/9906007
https://arxiv.org/abs/1109.6223
The more standard account is that somehow what happens at one detector affects what happens at the other. But even in that account, there is no prospect of knowing whether there is a beamsplitter at detectors 2,3 and 4 if the results of this process somehow agree with quantum mechanics.