Maxima of long memory stationary symmetric $\alpha$-stable processes, and self-similar processes with stationary max-increments
Takashi Owada
We derive a functional limit theorem for the partial maxima process based on a long memory stationary $\alpha$-stable process. The length of memory in the stable process is parameterized by a certain ergodic theoretical parameter in an integral representation of the process. The limiting process is no longer a classical extremal Fr\'echet process. It is a self-similar process with $\alpha$-Fr\'echet marginals, and it has the property of stationary max-increments, which we introduce in this paper. The functional limit theorem is established in the space $D[0,\infty)$ equipped with the Skorohod $M_1$-topology; in certain special cases the topology can be strengthened to the Skorohod $J_1$-topology.
1307.2356.tex
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\begin{document}

\bibliographystyle{ecta}

\title[Limit Theory for Partial Maxima]
{Maxima of long memory stationary symmetric $\alpha$-stable processes, 
  and self-similar processes with stationary max-increments}
\author{Takashi Owada}
\address{School of Operations Research and Information Engineering\\
Cornell University \\
Ithaca, NY 14853}
\email{to68@cornell.edu}

\author{Gennady Samorodnitsky}
\address{School of Operations Research and Information Engineering\\
and Department of Statistical Science \\
Cornell University \\
Ithaca, NY 14853}
\email{gs18@cornell.edu}


\thanks{This research was partially supported by the ARO
grants W911NF-07-1-0078 and W911NF-12-10385,  NSF grant  DMS-1005903
and NSA grant  H98230-11-1-0154  at Cornell University.}

\subjclass{Primary 60F17, 60G70. Secondary 37A40, 60G10, 60G52}
\keywords{Stable process, extreme value theory, conservative flow, pointwise dual ergodicity, sample maxima. \vspace{.5ex}}


\begin{abstract}
We derive a functional limit theorem for the partial maxima process
based on a long memory stationary $\alpha$-stable process. The length
of memory in the stable process is parameterized by a certain
ergodic theoretical parameter in an  integral representation of the
process. The limiting process is no longer a classical extremal Fr\'echet process. It is a self-similar process with $\alpha$-Fr\'echet
marginals, and it has the property of stationary max-increments, which
we introduce in this paper. The functional limit theorem is
established in the space $D[0,\infty)$ equipped with the Skorohod
$M_1$-topology; in certain special cases the topology can be
strengthened to the Skorohod $J_1$-topology. 
\end{abstract}

\maketitle

\section{Introduction} \label{sec:intro}

\baselineskip=18pt

The asymptotic behavior of the partial maxima sequence  $M_n=\max_{1 \leq
  k \leq n} X_k$, $n=1,2,\dots$ for an i.i.d. sequence
$(X_1,X_2,\ldots)$ of random
variables is the subject of the classical extreme value theory, dating
back to \cite{fisher:tippett:1928}. The basic result of this theory
says that only three one-dimensional distributions,  the Fr\'echet
distribution, the Weibull distribution and the Gumbel distribution,
have a max-domain of attraction. If $Y$ has one of these three
distributions, then for  a distribution in its domain 
of attraction, and a sequence of i.i.d. random variables with
that distribution, 
\begin{equation} \label{e:1din.conv}
\frac{M_n-b_n}{a_n} \Rightarrow Y\end{equation}
for properly chosen sequences $(a_n)$, $(b_n)$; see e.g. Chapter 1 in
\cite{resnick:1987} or Section 1.2 in
\cite{dehaan:ferreira:2006}. Under the same max-domain of attraction
assumption, a functional version of \eqref{e:1din.conv} was
established in \cite{lamperti:1964}: with the same sequences $(a_n)$,
$(b_n)$ as in \eqref{e:1din.conv},
\begin{equation} \label{e:funct.conv}
\left( \frac{M_{\lfloor nt\rfloor}-b_n}{a_n}, \ t\geq 0\right)
\Rightarrow (Y(t), \, t\geq 0)
\end{equation}
for a nondecreasing right continuous process $(Y(t), \, t\geq 0)$, and
the convergence is weak convergence in the Skorohod $J_1$ topology on
$D[0,\infty)$. The limiting process is often called {\it the extremal
  process}; its properties were established in
\cite{dwass:1964,dwass:1966} and \cite{resnick:rubinovitch:1973}. 

Much of the more recent research in extreme value theory concentrated
on the case when the underlying sequence $(X_1,X_2,\ldots)$ is
stationary, but may be dependent. In this case the extrema of the
sequence may cluster, and it is natural to expect that the limiting
results \eqref{e:1din.conv} and \eqref{e:funct.conv} will, in general,
have to be different. The extremes of moving average processes have
received special attention; see e.g. \cite{rootzen:1978},
\cite{davis:resnick:1985} and \cite{fasen:2005}.  The extremes of the 
GARCH(1,1) process were investigated in \cite{mikosch:starica:2000b}. 
The classical work on this subject of the extremes of dependent
sequences is 
\cite{leadbetter:lindgren:rootzen:1983}; in some cases this clustering
of the extremes can be characterized through the {\it extremal
  index} (introduced, originally, in \cite{leadbetter:1983}). The
latter is a number $0\leq \theta\leq 1$. Suppose that a stationary 
sequence $(X_1,X_2,\ldots)$ has this index, and let $(\tilde
X_1,\tilde X_2,\ldots)$ be an i.i.d. sequence with the same
one-dimensional marginal distributions as $(X_1,X_2,\ldots)$. If
\eqref{e:1din.conv} and \eqref{e:funct.conv} hold for the
i.i.d. sequence, then the corresponding limits will satisfy 
$\tilde Y\eid \tilde Y(1)$, but the limit in \eqref{e:1din.conv}
for the dependent sequence $(X_1,X_2,\ldots)$ will satisfy $Y \eid
\tilde Y(\theta)$. In particular, the limit will be equal to zero if
the extremal index is equal to zero. This case can be viewed as that
of long range dependence in the extremes, and it has been mostly
neglected by the extreme value community. Long range dependence is,
however, an important phenomenon in its own right, and in this paper
we take a step towards understanding how long range dependence affects
extremes. 


A random variable $X$ is said to have a regularly varying tail with
index $-\alpha$ for $\alpha>0$ if
$$
P(X>x) = x^{-\alpha} L(x), \ x>0\,,
$$
where $L$ is a slowly varying at infinity function, and the
distribution of any such random variable is in the max-domain of
attraction of the Fr\'echet distribution with the same parameter
$\alpha$; see e.g. \cite{resnick:1987}. Recall that the Fr\'echet law
$F_{\alpha,\sigma}$ on $(0,\infty)$ with the tail index $\alpha$ and scale $\sigma>0$
satisfies
\begin{equation} \label{e:frechet}
F_{\alpha,\sigma}(x) = \exp\bigl\{ -\sigma^\alpha x^{-\alpha}\bigr\}, \ x>0\,.
\end{equation}
Sometimes the term {\it $\alpha$-Fr\'echet} is used. 
In this paper we discuss the case of regularly varying tails and the resulting limits in
\eqref{e:funct.conv}. The limits obtained in this paper belong to
the family of the so-called {\it Fr\'echet processes}, defined
below. We would like to emphasize that, even for stationary sequences
with regularly varying tails, non-Fr\'echet limits may appear in
\eqref{e:funct.conv}. We are postponing a detailed discussion of this
point to a future publication. 

A stochastic process $(Y(t), \, t \in T)$ (on an arbitrary parameter
space $T$) is called a Fr\'echet process if for all $n \geq 1$, $a_1,
\dots, a_n >0$ and $t_1, \dots, t_n \in T$, the weighted maximum
$\max_{1 \leq j \leq n} a_j Y(t_j)$ follows a Fr\'echet law as in
\eqref{e:frechet}. The best known Fr\'echet process is the extremal
Fr\'echet process obtained in the scheme \eqref{e:funct.conv} starting
with an i.i.d. sequence with regularly varying tails. The extremal
Fr\'echet process $\bigl( Y(t),\, t\geq 0\bigr)$ has finite
dimensional distributions defined by 
\begin{equation} \label{e:extreme.frechet}
\bigl( Y(t_1),Y(t_2),\ldots, Y(t_n)\bigr) \eid \biggl(
X^{(1)}_{\alpha,t_1^{1/\alpha}}, \ \max \Bigl(
X^{(1)}_{\alpha,t_1^{1/\alpha}}, \,
X^{(2)}_{\alpha,(t_2-t_1)^{1/\alpha}}\Bigr), \, \ldots \, 
\end{equation}
$$
\max \Bigl(
X^{(1)}_{\alpha,t_1^{1/\alpha}}, \,
X^{(2)}_{\alpha,(t_2-t_1)^{1/\alpha}}, \ldots,
X^{(n)}_{\alpha,(t_n-t_{n-1})^{1/\alpha}}\Bigr)\biggr)
$$
for all $n$ and $0\leq t_1<t_2:\ldots <t_n$. The different random
variables in the right hand side of \eqref{e:extreme.frechet} are
independent, with $X^{(k)}_{\alpha,\sigma}$ having the  Fr\'echet law
$F_{\alpha,\sigma}$ in \eqref{e:frechet}, for any $k=1,\ldots, n$. This
independence makes  the extremal Fr\'echet processes the extremal
Fr\'echet versions of the better known L\'evy processes. 
The structure of general  Fr\'echet processes has been extensively
studied in the last several years. These processes were introduced in
\cite{stoev:taqqu:2005}, and their representations (as a part of a
much more general context) were studied in
\cite{kabluchko:stoev:2012}. Stationary Fr\'echet
processes (in particular, their ergodicity and mixing) were discussed
in \cite{stoev:2008}, \cite{kabluchko:schlather:dehaan:2009} and 
\cite{wang:stoev:2010}. 

In this paper we concentrate on the maxima of stationary
$\alpha$-stable processes with $0<\alpha<2$. Recall that a random
vector $\BX$ in $\bbr^d$ is called $\alpha$-stable if for any $A$ and
$B>0$ we have
$$
A\BX^{(1)} + B\BX^{(2)} \eid (A^\alpha+B^\alpha)^{1/\alpha} \BX+ \by\,,
$$
where $\BX^{(1)}$ and  $\BX^{(2)}$ are i.i.d. copies of $\BX$, and
$\by$ is a deterministic vector (unless $\BX$ is deterministic,
necessarily, $0<\alpha\leq 2$). A stochastic process $(X(t), \, t \in
T)$ is called $\alpha$-stable if all of its finite-dimensional
distributions are $\alpha$-stable. We refer the reader to
\cite{samorodnitsky:taqqu:1994} for information on $\alpha$-stable
processes. When $\alpha=2$, an $\alpha$-stable process is Gaussian,
while in the case $0<\alpha<2$, both the left and the right tails of a
(nondegenerate) $\alpha$-stable random variable $X$ are (generally)
regularly varying with exponent $\alpha$. That is, 
$$
P(X>x) \sim c_+ x^{-\alpha}, \ \ P(X<-x) \sim c_- x^{-\alpha} \ \
\text{as $x\to\infty$}
$$
for some $c_+, \, c_-\geq 0$, $c_++c_->0$. That is, if
$(X_1,X_2,\ldots)$ is an i.i.d. sequence of $\alpha$-stable random
variables, then the i.i.d. sequence  $(|X_1|,|X_2|,\ldots)$ satisfies
\eqref{e:1din.conv} and \eqref{e:funct.conv} with $a_n=n^{1/\alpha}$
(and $b_n=0$), $n\geq 1$. Of course, we are not planning to study the
extrema of an i.i.d. $\alpha$-stable sequence. Instead, we will study
the maxima of (the absolute values of) a stationary $\alpha$-stable
process. The reason we have chosen to work with stationary $\alpha$-stable
processes is that their structure is very rich, and is also relatively
well understood. This will allow us to study the effect of that
structure on the limit theorems \eqref{e:1din.conv} and
\eqref{e:funct.conv}. We are specifically interested in the long range
dependent case, corresponding to the zero value of the extremal
index. 

The structure of stationary symmetric $\alpha$-stable (\SaS) processes
has been 
clarified in the last several years in the works of Jan Rosi\'nski;
see e.g. \cite{rosinski:1995, rosinski:2006}. The integral
representation of 
such a process can be chosen to have a very special form. The class of
stationary \SaS\ processes we will investigate requires a 
representation slightly more restrictive than the one generally
allowed. Specifically, we will consider processes of the form 
\begin{equation}  \label{e:underlying.proc}
X_n = \int_E f \circ T^n(x) \, dM(x), \ \ \ n=1,2,\dots\,,
\end{equation}
where $M$ is a \SaS\ random measure on a measurable space
$(E,\mathcal{E})$ with a $\sigma$-finite {\it infinite} control
measure $\mu$. The map $T:E
\to E$ is a measurable map that preserves the measure $\mu$. Further,
$f \in L^{\alpha}(\mu)$. See
\cite{samorodnitsky:taqqu:1994} for details on $\alpha$-stable random
measures and integrals with respect to these measures. It is
elementary to check that  a process with a 
representation \eqref{e:underlying.proc} is, automatically,
stationary. Recall that any stationary \SaS\ process has a
representation of the form: 
 \begin{equation}  \label{e:general.integral}
X_n = \int_E f_n(x) dM(x), \ \ \ n=1,2,\dots\,,
\end{equation}
with 
\begin{equation}  \label{e:general.rosinski}
f_n(x) = a_n(x) \left( \frac{d\mu \circ T^{n}}{d\mu}(x) \right)^{1/\alpha} f\circ T^{n}(x),  \ \ \ x \in E
\end{equation}
for $n=1,2,\dots$, where $T:E \to E$ is a one-to-one map with both $T$
and $T^{-1}$ measurable, mapping the control measure $\mu$ into an
equivalent measure, and  
$$
a_n(x) = \prod_{j=0}^{n-1} u \circ T^j(x), \ \ \ x \in E
$$
for $n=1,2,\dots$, with $u:E \to \{-1,1  \}$ a measurable
function. Here $M$ is \SaS\ (and $f \in L^{\alpha}(\mu)$). See
\cite{rosinski:1995}. The main restriction of
\eqref{e:underlying.proc} is, therefore, to a measure preserving
map $T$.

It has been observed that the ergodic-theoretical properties of the
map $T$, either in \eqref{e:underlying.proc} or in
\eqref{e:general.rosinski}, have a major impact on the memory of a
stationary $\alpha$-stable process. See
e.g. \cite{surgailis:rosinski:mandrekar:cambanis:1993},
\cite{samorodnitsky:2004a, samorodnitsky:2005}, \cite{roy:2008},
\cite{resnick:samorodnitsky:2004}, \cite{owada:samorodnitsky:2012}, 
\cite{owada:2013}. The
most relevant for this work is the result of
\cite{samorodnitsky:2004a}, who proved  that,
if the map $T$ in \eqref{e:underlying.proc} or in 
\eqref{e:general.rosinski} is conservative, then using the
normalization $a_n=n^{1/\alpha}$ ($b_n=0$) in \eqref{e:1din.conv}, as
indicated by the marginal tails, produces the zero limit, so the
partial maxima grow, in this case, strictly slower than at the rate of
$n^{1/\alpha}$. On the other hand, if the map $T$ is not conservative,
then the normalization $a_n=n^{1/\alpha}$ in \eqref{e:1din.conv} is
the correct one, and it leads to a Fr\'echet limit (we will survey
the ergodic-theoretical notions in the next section). Therefore, the
extrema of \SaS\ processes corresponding to conservative flows cluster
so much that the sequence of the partial maxima grows at a slower rate
than that indicated by the marginal tails. This case can be thought of
as indicating long range dependence. It is, clearly, inconsistent with
a positive extremal index. 

The Fr\'echet limit obtained in \eqref{e:1din.conv} by
\cite{samorodnitsky:2004a} remains valid when the map $T$ is
conservative (but with the normalization of a smaller order than
$n^{1/\alpha}$), as long as the map $T$ satisfies a certain additional
assumption. If one views the stationary $\alpha$-stable process as
a natural function of the Poisson points forming the random measure
$M$ in \eqref{e:general.integral} then, informally, this assumption
guarantees that only the largest Poisson point contributes,
distributionally, to the asymptotic behavior of the partial maxima of
the process. In this paper we restrict ourselves to this situation as
well. However, we will look at the limits obtained in the
much more informative functional scheme \eqref{e:funct.conv}. In this
paper the assumption on the map $T$ will be expressed in terms of the
rate of growth of the so-called return sequence, which we define in
the sequel. We would like to emphasize that, when this return sequence
grows at a rate slower than the one assumed in this paper, new
phenomena seem to arise. Multiple Poisson points may contribute to the
asymptotic distribution of the partial maxima, and non-Fr\'echet limit
may appear in \eqref{e:funct.conv}. We leave a detailed study of this to a
subsequent work. 

In the next section we provide the elements of the infinite ergodic
theory needed for the rest of the paper. In Section
\ref{sec:lim.processes} we introduce a new notion, that of a process
with stationary max-increments. It turns out that the possible limits
in the functional maxima scheme \eqref{e:funct.conv} (with $b_n=0$)
are self-similar with stationary max-increments. We discuss the
general properties of such processes and then specialize to the
concrete limiting process we obtain in the main result of the paper,
stated and proved in Section \ref{sec:FLTPM}. 




\section{Ergodic Theoretical Notions} \label{sec:ergodic}

In this section we present some basic notation and notions of, mostly infinite, ergodic theory used in
the sequel. The main references are \cite{krengel:1985},
\cite{aaronson:1997}, and \cite{zweimuller:2009}.  

Let $(E,\mathcal{E},\mu)$ be a $\sigma$-finite, infinite measure
space. We will say that $A = B$ mod $\mu$ if $A, B \in \mathcal{E}$
and $\mu(A\triangle B)=0$. For $f\in L^1(\mu)$ we will often write
$\mu(f)$ for the integral $\int f\, d\mu$. 

Let $T:E \to E$ be a measurable map preserving the measure $\mu$. The
sequence $(T^n)$ of iterates 
of $T$ is called a {\it flow}, and the ergodic-theoretical properties of
the map and the flow are identified. A map $T$ is called {\it ergodic} if
any $T$-invariant set $A$   (i.e., a set such that $T^{-1}A = A$ mod
$\mu$) satisfies $\mu(A)=0$ or $\mu(A^c)=0$. A map $T$ is said to be
{\it conservative} if  
$$
\sum_{n=1}^{\infty} \one_A \circ T^n = \infty \ \ \text{a.e. on } A
$$
for any $A \in \mathcal{E}$, $0 < \mu(A) < \infty$; if $T$ is also
ergodic, then the restriction ``{\it on} $A$'' is not needed. 

The {\it conservative part} of a measure-preserving $T$ is the largest
$T$-invariant subset $C$ of $E$ such that the restriction of $T$ to
$C$ is conservative. The set $D=E \setminus C$ is the  {\it
  dissipative part} of $T$ (and the decomposition $E=C\cup D$ is
called {\it the Hopf decomposition} of $T$). 
 
The  {\it dual operator} $\widehat{T}: L^1(\mu) \to L^1(\mu)$ is
defined by 
\begin{equation} \label{e:define.dual}
\widehat{T} f = \frac{d \bigl(\nu_f \circ T^{-1}\bigr)}{d \mu}, \ f\in L^1(\mu)\,,
\end{equation}
where $\nu_f$ is the signed measure  $\nu_f(A) = \int_{A} f
d\mu$, $A \in \mathcal{E}$. The dual operator satisfies the  duality
relation  
\begin{equation} \label{e:dual.rel}
\int_E   \widehat{T} f\cdot g\, d\mu = \int_E f\cdot g \circ T\, d\mu
\end{equation}
for $f\in L^1(\mu), \, g\in L^\infty(\mu)$. Note that
\eqref{e:define.dual} makes sense for  any nonnegative measurable
function $f$ on $E$, and the resulting  $\widehat{T} f$ is again a nonnegative measurable
function. Furthermore, \eqref{e:dual.rel} holds for arbitrary
nonnegative measurable functions $f$ and $g$.  

A conservative, ergodic and measure preserving map $T$ is said to be
{\it pointwise dual ergodic}, if there exists a normalizing sequence
$a_n \nearrow \infty$ such that  
\begin{equation}  \label{e:pde}
\frac{1}{a_n} \sum_{k=1}^n \widehat{T}^k f \to \mu(f) \ \ \ \text{a.e.} \; \text{for every } f \in L^1(\mu)\,. 
\end{equation}
The property of pointwise dual ergodicity rules out invertibility of
the map $T$. 

Sometimes we require that for some functions the above convergence
takes place uniformly on a certain set. A set $A \in
\mathcal{E}$ with $0 < \mu(A) < \infty$ is said to be a {\it
  uniform set} for a conservative, ergodic and measure preserving map
$T$, if there exist a normalizing sequence $a_n \nearrow \infty$ and a
nontrivial nonnegative measurable function $f \in L^1(\mu)$
(nontriviality means that $f$ is different from zero on a set of
positive measure) such that  
\begin{equation}  \label{e:uniform_set}
\frac{1}{a_n} \sum_{k=1}^n \widehat{T}^k f \to \mu(f) \ \ \ \text{uniformly, a.e. on } A\,. 
\end{equation}
If  \eqref{e:uniform_set} holds for $f=\one_A$, the set $A$ is  called
{\it a Darling-Kac set}. A conservative, ergodic and measure
preserving map $T$ is pointwise dual ergodic if and only if $T$ admits
a uniform set; see Proposition 3.7.5 in \cite{aaronson:1997}. In
particular, it is legitimate to use the same normalizing 
sequence $(a_n)$ both in (\ref{e:pde}) and (\ref{e:uniform_set}). 

Let $A \in \mathcal{E}$ with $0 < \mu(A) < \infty$. The frequency of
visits to the set $A$ along the trajectory $( T^n x), \, x\in E$, is
naturally related to the  {\it wandering rate} sequence 
\begin{equation}  \label{e:wanderingrate}
w_n = \mu \Bigl(\bigcup_{k=0}^{n-1} T^{-k}A\Bigr)\,. 
\end{equation}
If we  define the first entrance time to $A$ by 
$$
\varphi_A(x) = \min \{ n \geq 1: T^n x \in A \}
$$
(notice that $\varphi_A < \infty$ a.e. on $E$ since $T$ is conservative and
ergodic), then $w_n \sim \mu(\varphi_A < n)$ as $n\to\infty$. Since $T$
is also measure preserving, we have 
  $\mu(A \cap \{ \varphi_A > k \}) = \mu(A^c \cap \{ \varphi_A = k \})$
  for $k \geq 1$. Therefore, alternative expressions for the wandering rate 
  sequence are 
$$
w_n = \mu(A) + \sum_{k=1}^{n-1} \mu (A^c \cap \{ \varphi_A = k  \}) = \sum_{k=0}^{n-1} \mu(A \cap \{ \varphi_A > k \})\,.
$$
 
Suppose now that $T$ is a pointwise dual ergodic map, and let $A$ be a
uniform set for $T$. It turns out that, under an assumption of regular
variation, there is a precise connection between the wandering rate sequence $(w_n)$ and the normalizing sequence
$(a_n)$ in \eqref{e:pde} and \eqref{e:uniform_set}. Specifically, let
$RV_{\gamma}$ represent the class of regularly varying at infinity sequences (or
functions, depending on the context) of
index $\gamma$. If either $(w_n) \in RV_{\beta}$ or $(a_n) \in RV_{1-\beta}$ for some
$\beta \in [0,1]$, then  
\begin{equation}  \label{e:prop3.8.7} 
a_n \sim \frac{1}{\Gamma(2-\beta) \Gamma(1+\beta)} \frac{n}{w_n} \ \ \
\text{as } n \to \infty\,. 
\end{equation}
Proposition 3.8.7 in \cite{aaronson:1997} gives one direction of this statement, but the argument is easily reversed. 

\bigskip

We finish this section with a statement on distributional convergence
of the partial maxima for pointwise dual ergodic flows. It will be
used repeatedly in the proof of the main theorem. For a measurable
function $f$ on $E$ define 
$$
M_n(f)(x) = \max_{1 \leq k \leq n}|f \circ T^k(x)|\,, \ \ x \in E, \ n \geq 1\,.
$$
In the sequel we will use the convention $\max_{k\in K}b_k = 0$ for
a nonnegative sequence $(b_n)$, if $K=\emptyset$. 

\begin{proposition}  \label{p:max.ergodic}
Let $T$ be a pointwise dual ergodic map on a $\sigma$-finite,
infinite, measure space $(E,\mathcal{E},\mu)$. We assume that the
normalizing sequence $(a_n)$ is regularly varying with exponent
$1-\beta$ for some $0 < \beta \leq 1$. Let $A \in \mathcal{E}$, $0 <
\mu(A) < \infty$, be a uniform set for $T$. Define a probability
measure on $E$ by $\mu_n (\cdot) = \mu(\cdot \cap \{\varphi_A \leq n\})
/ \mu(\{\varphi_A \leq n \})$.  
Let $f:E \to \bbr$ be a measurable bounded function supported by the
set $A$. Let $\| f \|_{\infty} = \inf \{M : |f(x)| \leq M \ \text{a.e. on } A  \}$. Then
$$
 \left( M_{\lfloor nt\rfloor}(f), \ 0\leq t\leq 1 \right)
 \Rightarrow  \|f\|_\infty \left( \one_{\{ V_{\beta} \leq t \}}, \ 0\leq t\leq 1 
 \right) \ \ \ \text{in the $M_1$ topology on $D[0,1] $,}
$$
where the law of the left hand side is computed with respect to
$\mu_n$, and $V_{\beta}$ is a random variable defined on a probability
space $(\Omega^{\prime},\mathcal{F}^{\prime},P^{\prime})$ with
$P^{\prime}(V_{\beta} \leq x) = x^{\beta}$, $0 < x \leq 1$.  
\end{proposition}
\begin{proof}
For $0<\vep<1$ let $A_\vep = \{ x\in A:\, |f(x)|\geq (1-\vep)\|
f\|_\infty\}$. Note that each $A_\vep$ is uniform since $A$ is
uniform. Clearly, 
$$
(1-\vep)\| f\|_\infty \one_{ \{\varphi_{A_\vep}(x)\leq nt \} } \leq 
M_{\lfloor nt\rfloor}(f)(x)\leq \| f\|_\infty \one_{ \{\varphi_A(x)\leq nt \} }
 \ \ \ \mu \text{-a.e.} 
$$
for all $n\geq 1$ and $0\leq t\leq 1$. Since for monotone functions
weak convergence in the $M_1$ topology is implied by  convergence in
finite dimensional distributions, we can use Theorem 3.2 in
\cite{billingsley:1999} in a finite-dimensional situation. The
statement of the proposition will follow once we show that, for a
uniform set $B$ (which could be either $A$ or $A_\vep$) the law of
$\varphi_B/n$ under $\mu_n$ converges to the law of $V_{\beta}$. Let
$(w_n^{(B)})$ be the corresponding  wandering rate sequence. Since
\eqref{e:prop3.8.7} holds for $(w_n^{(B)})$ with  the same normalizing
constants $a_n$, 
$$
\mu_n \left( \frac{\varphi_{B}}{n} \leq x \right) =  \frac{\mu(\varphi_{B}
  \leq \lfloor nx \rfloor )}{\mu(\varphi_A \leq n)} \sim
\frac{w_{\lfloor nx \rfloor}^{(B)}}{w_n} \to x^{\beta}  
$$
for all $0 < x \leq 1$, because the wandering rate sequence is regularly
varying with index $\beta$ by \eqref{e:prop3.8.7}. 
\end{proof}


\section{Self-similar processes with stationary
 max-increments} \label{sec:lim.processes}


The limiting process obtained in the next section shares with any possible
limits  in the functional maxima scheme \eqref{e:funct.conv} (with
$b_n=0$) two very specific properties, one of which is classical, and
the other is less so. Recall that a stochastic process $\bigl( Y(t),
\, t\geq 0\bigr)$ is called self-similar with exponent $H$ of
self-similarity if for any $c>0$
$$
\bigl( Y(ct), \, t\geq 0\bigr)\eid \bigl( c^HY(t), \, t\geq 0\bigr)
$$
in the sense of equality of all finite-dimensional distributions. The
best known classes of self-similar processes arise in various versions
of a functional central limit theorem for stationary processes, and
they have an additional property of stationary increments. Recall that
a stochastic process $\bigl( Y(t), \, t\geq 0\bigr)$ is said to have
stationary increments if for any $r\geq 0$
\begin{equation} \label{e:stat.incr} 
\bigl( Y(t+r)-Y(r), \, t\geq 0\bigr) \eid \bigl( Y(t)-Y(0), \, t\geq
0\bigr) \,;
\end{equation}
see e.g. \cite{embrechts:maejima:2002} and
\cite{samorodnitsky:2006LRD}. In the context of the functional limit
theorem for the maxima \eqref{e:funct.conv}, a different property
appears. 
\begin{definition} \label{d:max.stat.def}
A stochastic process $( Y(t), \, t \geq 0 )$ is said to have
stationary max-increments if for every $r \geq 0$, there exists,
perhaps on an enlarged probability space, a
stochastic process $\bigl( Y^{(r)}(t), \, t \geq 0 \bigr)$ such that 
\begin{align} \label{e:def.statmaxi}
\bigl( Y^{(r)}(t), \, t \geq 0 \bigr) &\stackrel{d}{=} ( Y(t), \, t
\geq 0 )\,, \nonumber \\
( Y(t+r), \, t \geq 0 ) &\stackrel{d}{=} \bigl( Y(r) \vee Y^{(r)}(t), \, t \geq 0 \bigr)\,.
\end{align}
\end{definition}

Notice the analogy between the definition \eqref{e:stat.incr}  of
stationary increments (when $Y(0)=0$) and Definition
\ref{d:max.stat.def}. Since the operations of taking the maximum is not
invertible (unlike summation), the latter definition, by necessity, is
stated in terms of existence of the max-increment process $\bigl(
Y^{(r)}(t), \, t \geq 0 \bigr)$. 

\begin{theorem} \label{t:max.lamperti}
Let $(X_1,X_2,\ldots)$ be a stationary sequence. Assume that for some
sequence $a_n\to\infty$, and a stochastic process $(Y(t), \, t\geq 0)$
such that, for every $t>0$, $Y(t)$ is not a constant, 
$$
\left( \frac{1}{a_n} M_{\lfloor nt\rfloor}, \ t\geq 0\right)
\Rightarrow (Y(t), \, t\geq 0)
$$
in terms of convergence of finite-dimensional distributions. Then
$(Y(t), \, t\geq 0)$ is self-similar with exponent $H>0$ of
self-similarity, and has stationary max-increments. Furthermore, the
sequence $(a_n)$ is regularly varying with index $H$. 
\end{theorem}
\begin{proof}
The facts that the limiting process $(Y(t), \, t\geq 0)$ is self-similar with exponent $H>0$ of
self-similarity, and that the
sequence $(a_n)$ is regularly varying with index $H$, follow from the
Lamperti theorem; see \cite{lamperti:1962}, or Theorem 2.1.1 in
\cite{embrechts:maejima:2002}. Note that the distinction between
convergence along a discrete sequence $n\to\infty$, or along a continuous
sequence $\lambda\to\infty$, disappears for maxima of stationary
processes.

We check now the  stationarity of the max-increments of the limiting
process. Let $r>0$, and $t_i>0, \, i=1,\ldots, k$, some $k\geq 1$. 
Write 
\begin{equation} \label{e:decompose}
\frac{1}{a_n} M_{\lfloor n(t_i+r)\rfloor} = \frac{1}{a_n} M_{\lfloor
  nr\rfloor} \bigvee \frac{1}{a_n} \max_{nr<j\leq n(t_i+r)} X_j, \
i=1,\ldots, k\,.
\end{equation} 
By the assumption of the theorem and stationarity of the process
$(X_1,X_2,\ldots)$, 
$$
\frac{1}{a_n} M_{\lfloor nr\rfloor} \Rightarrow Y(r), \ \ 
\Bigl( \frac{1}{a_n} \max_{nr<j\leq n(t_i+r)} X_j, \
i=1,\ldots, k\Bigr) \Rightarrow \bigl( Y(t_1),\ldots, Y(t_k)\bigr)
$$
as $n\to\infty$. Using the standard tightness argument, we conclude
that for every sequence $n_m\to\infty$ there is a subsequence $n_{m(l)}\to\infty$ such that
$$
\biggl( \frac{1}{a_{n_{m(l)}}} M_{\lfloor {n_{m(l)}}r\rfloor}, \ 
\Bigl( \frac{1}{a_{n_{m(l)}}} \max_{n_{m(l)}r<j\leq n_{m(l)}(t_i+r)} X_j, \
i=1,\ldots, k\Bigr)\biggr)  \Rightarrow \Bigl( Y(r), \, \bigl( Y^{(r)}(t_1),\ldots, Y^{(r)}(t_k)\bigr)\Bigr)
$$
as $l\to\infty$, where $\bigl( Y^{(r)}(t_1),\ldots,
Y^{(r)}(t_k)\bigr)\eid \bigl( Y(t_1),\ldots, Y(t_k)\bigr)$. 

Let now $\tau_i, \, i=1,2,\ldots$ be an enumeration of the rational
numbers in $[0,\infty)$. A diagonalization argument shows that there
is a sequence $n_m\to\infty$ and a stochastic process $\bigl(
Y^{(r)}(\tau_i), \, i=1,2,\dots \bigr)$ with $\bigl( Y^{(r)}(\tau_i), \, i=1,2,\dots
\bigr)\eid \bigl( Y(\tau_i), \, i=1,2,\dots \bigr)$ such that 
\begin{equation} \label{e:rational.conv}
\biggl( \frac{1}{a_{n_{m}}} M_{\lfloor {n_{m}}r\rfloor}, \ 
\Bigl( \frac{1}{a_{n_{m}}} \max_{n_{m}r<j\leq n_{m}(\tau_i+r)} X_j, \
i=1,2,\ldots \Bigr)\biggr)  \Rightarrow \Bigl( Y(r), \, \bigl(
Y^{(r)}(\tau_i), \ i=1,2,\ldots\bigr)\Bigr)
\end{equation}
in finite-dimensional distributions, as $m\to\infty$. We extend the
process $Y^{(r)}$ to the entire positive half-line by setting
$$
Y^{(r)}(t) = \frac12 \Bigl( \lim_{\tau\uparrow t, \ \text{rational}}
Y^{(r)}(\tau)  + \lim_{\tau\downarrow t, \ \text{rational}}
Y^{(r)}(\tau)  \Bigr), \ \ t\geq 0\,.
$$
Then \eqref{e:rational.conv} and monotonicity imply that as $m \to \infty$,
\begin{equation} \label{e:full.conv}
\biggl( \frac{1}{a_{n_{m}}} M_{\lfloor {n_{m}}r\rfloor}, \ 
\Bigl( \frac{1}{a_{n_{m}}} \max_{n_{m}r<j\leq n_{m}(t+r)} X_j, \
t\geq 0 \Bigr)\biggr)  \Rightarrow \Bigl( Y(r), \, \bigl(
Y^{(r)}(t), \ t\geq 0\bigr)\Bigr)
\end{equation}
in finite-dimensional distributions. Now the stationarity of max-increments follows from \eqref{e:decompose},
\eqref{e:full.conv} and continuous mapping theorem. 
\end{proof}

\begin{remark} \label{rk:sup.measures}
{\rm
Self-similar processes with stationary max-increments arising in a 
functional  maxima scheme \eqref{e:funct.conv} are close in spirit to
the stationary self-similar extremal processes of
\cite{obrien:torfs:vervaat:1990}, while extremal processes themselves
are defined as random sup measures. A
random sup measure is, as its name implies, indexed by sets. They also
arise in a limiting maxima scheme similar to \eqref{e:funct.conv}, but
with a stronger notion of convergence. Every stationary self-similar
extremal processes trivially produces a self-similar process with
stationary max-increments via restriction to sets of the type $[0,t]$
for $t\geq 0$, but the connection between the two objects remains
unclear. Our limiting process in Theorem
\ref{t:main.max} below can be extended to a stationary self-similar
extremal processes, but the extension is highly nontrivial, and will
not be pursued here. 
}
\end{remark}


It is not our goal in this paper to study in details the properties of
self-similar processes with stationary max-increments, so we restrict
ourselves to the following basic result,
\begin{proposition} \label{pr:sssmaxi}
Let $\bigl( Y(t), \, t \geq 0 \bigr)$ be a nonnegative self-similar
process with 
stationary max-increments, and exponent $H$ of self-similarity. Suppose $\bigl(Y(t), \, t \geq 0 \bigr)$ is not identically zero. Then
$H\geq 0$, and the following statements hold.

{\rm (a)} \ If $H=0$, then $Y(t)=Y(1)$ a.s. for every $t>0$.

{\rm (b)} \ If $0<EY(1)^p<\infty$ for some $p>0$, then $H\leq 1/p$.

{\rm (c)} If $H>0$, $\bigl( Y(t), \, t \geq 0 \bigr)$ is continuous in probability.
\end{proposition}
\begin{proof}
By the stationarity of max-increments, $Y(t)$ is stochastically
increasing with $t$. This implies that $H\geq 0$.

If $H=0$, then $Y(n)\eid Y(1)$ for each $n=1,2\ldots$. We use
\eqref{e:def.statmaxi} with $r=1$. Using $t=1$ we see that, in the
right hand side of \eqref{e:def.statmaxi}, $Y(1)=Y^{(1)}(1)$
a.s. Since $Y^{(1)}(n)\geq Y^{(1)}(1)$ a.s., we conclude, using $t=n$ in the
right hand side of \eqref{e:def.statmaxi}, that $Y(1)=Y^{(1)}(n)$ a.s for each $n=1,2,\ldots$. By monotonicity, we
conclude that the process $\bigl( Y^{(1)}(t), \, t \geq 0 \bigr)$,
hence also the process $\bigl( Y(t), \, t \geq 0 \bigr)$, 
is a.s. constant on $[1,\infty)$ and then, by self-similarity, also on
$(0,\infty)$. 

Next, let $p>0$ be such that $0<EY(1)^p<\infty$. It follows from
\eqref{e:def.statmaxi} with $r=1$ that
$$
2^H Y(1)\eid Y(2)\eid \max\bigl( Y(1), \, Y^{(1)}(1)\bigr)\,.
$$
Therefore, 
$$
2^{pH} EY(1)^p = E Y(2)^p = E \bigl[ Y(1)^p \vee Y^{(1)}(1)^p \bigr] \leq 2 EY(1)^p
$$
This means that $pH\leq 1$. 

Finally, we take arbitrary $0<s<t$. We  use \eqref{e:def.statmaxi} with $r=s$. For every $\eta > 0$, 
$$
P \bigl( Y(t) - Y(s) > \eta \bigr) = P \bigl( Y(s) \vee Y^{(s)}(t-s) - Y(s) > \eta \bigr)
$$
$$
\leq P \bigl(  Y^{(s)}(t-s) > \eta \bigr) = P \bigl( (t-s)^H Y(1) >
\eta \bigr). 
$$
Hence continuity in probability. 
\end{proof}


We now define the limiting process obtained in  the main limit theorem
of Section \ref{sec:FLTPM}, and place it in the general framework
introduced earlier in this section. Let $\alpha>0$, and consider
the extremal Fr\'echet process  $Z_{\alpha}(t), \, t\geq 0$, 
defined in \eqref{e:extreme.frechet}, with the scale $\sigma=1$. For
$0<\beta<1$ we define a new stochastic process by  
$$
Z_{\alpha, \beta}(t) =  Z_{\alpha}(t^\beta), \ t\geq 0\,.
$$
Since the extremal Fr\'echet process is self-similar with
$H=1/\alpha$, it is immediately seen that the process $Z_{\alpha,
  \beta}$ is self-similar with $H=\beta/\alpha$.

 We claim that the process $Z_{\alpha,  \beta}$ has stationary
 max-increments as well. To show this, we start with a useful
 representation of the extremal Fr\'echet process  $Z_{\alpha}(t), \,
 t\geq 0$ in terms of the points of a Poisson random measure. Let 
$\bigl( (j_k,s_k)\bigr)$ be the points of a Poisson random measure on $\bbr_+^2$
with mean measure $\rho_{\alpha} \times \lambda$, where 
$\rho_{\alpha}(x,\infty) = x^{-\alpha}$, $x>0$ and $\lambda$ is the
Lebesgue measure on $\bbr_+$.  Then an elementary calculation shows
that 
$$
\bigl( Z_{\alpha}(t), \,
 t\geq 0\bigr) \eid  \Bigl( \sup\bigl\{ j_k:\, s_k\leq t\bigr\}, \,
 t\geq 0\Bigr)\,. 
$$
Therefore, $\bigl( Z_{\alpha,\beta}(t), \,  t\geq 0\bigr) \eid \bigl(
U_{\alpha,\beta}(t), \,  t\geq 0\bigr) $, where 
\begin{equation} \label{e:represent.U}
U_{\alpha,\beta}(t) = \sup \, \{ \, j_k : s_k \leq t^{\beta} \}\,, \ \ t \geq 0\,.
\end{equation}

Given $r > 0$, we define 
$$
U_{\alpha,\beta}^{(r)}(t) = \sup \, \{ \, j_k : (t+r)^{\beta}-t^{\beta} \leq s_k \leq (t+r)^{\beta} \}\,.
$$
Since 
$$
\bigl( (t_1+r)^{\beta}-t_1^{\beta}, \, (t_1+r)^{\beta}\bigr) \subset 
\bigl( (t_2+r)^{\beta}-t_2^{\beta}, \, (t_2+r)^{\beta}\bigr)
$$
for $0\leq t_1<t_2$, it follows that 
$$
\bigl( U_{\alpha,\beta}^{(r)}(t), \, t \geq 0 \bigr) \stackrel{d}{=}
\bigl( U_{\alpha,\beta}(t), \, t \geq 0 \bigr)\,. 
$$
Furthermore, since $(t+r)^{\beta}-t^{\beta}\leq r^\beta$, we see that 
$$
U_{\alpha,\beta}(t+r) = U_{\alpha,\beta}(r) \vee U_{\alpha,\beta}^{(r)}(t) \ \ \text{for all } t \geq 0\,.
$$
This means that the process $U_{\alpha,\beta}$ has stationary
max-increments and, hence, so does the process $Z_{\alpha,\beta}$. 
 
It is interesting to note that, by part (b) of Proposition
\ref{pr:sssmaxi}, any $H$-self-similar process with stationary
max-increments and $\alpha$-Fr\'echet marginals, must satisfy $H\leq
1/\alpha$. The exponent $H=\beta/\alpha$ with $0<\beta\leq 1$ of the
process $Z_{\alpha,\beta}$ (with $\beta=1$ corresponding to the
extremal Fr\'echet process $Z_{\alpha}$) covers the entire interval
$(0,1/\alpha]$. Therefore, the upper bound of part (b) of Proposition
\ref{pr:sssmaxi} is, in general, the best possible. 

As a general phenomenon, the stationary increments of a self-similar
stochastic process arising in a functional central limit theorem (for
partial sums) exhibit dependence reflecting, at least partially the
dependence in the original stationary process. This can be seen, for
example, in  \cite{owada:samorodnitsky:2012} that starts with a
stationary process similar in its nature to the one considered in this
paper. Do the stationary max-increments of a process arising in the
functional maxima scheme \eqref{e:funct.conv} reflect similarly the
dependence structure in the original process? 

The fact the operation of taking partial maxima tends to reduce the
memory in the original process has been noticed before; for example,
for stationary Gaussian processes partial maxima converge, with the
same normalization, to the extremal Gumbel process, as long as the
correlations decay more rapidly than $(\log n)^{-1}$; see
\cite{berman:1964}. Within that range of correlation decay, on the
other hand, one would observe that the partial sums stop converging to
the Brownian motion, and start converging to the Fractional Brownian
motion, and the normalization will change as well. A related
 example with power tails is given by the infinite moving average
 models of the form $X_n = 
\sum_{j=0}^{\infty} c_j Z_{n-j}$, where $(Z_n)$ are i.i.d. random 
variables, belonging to the max-domain of attraction of an
$\alpha$-Fr\'echet law, and $(c_j)$ is a deterministic sequence
satisfying certain conditions for the process to be well-defined. In
this case it was shown by \cite{davis:resnick:1985} that the
normalized partial maxima of $(X_n)$  converge in the functional sense
to an extremal $\alpha$-Fr\'echet process; this paper assumes absolute
summability of $(c_j)$, but it is only needed for the tail estimate
(2.7), which is valid without the absolute summability, see
\cite{mikosch:samorodnitsky:2000}. In contrast, partial sums of 
the infinite moving average processes 
can converge, if $0<\alpha<2$, in certain cases (some of which prevent
lack of summability of the 
coefficients), to the linear fractional $\alpha$-stable motions, and
not to the $\alpha$-stable L\'evy motions; see e.g. Section 4.7 in
\cite{whitt:2002}.  

With the discussion in view, we investigate the dependence properties
of the stationary max-increments of the process $Z_{\alpha,\beta}$
defined above. This process will appear, in the case $0< \alpha < 2$
and $1/2 < \beta < 1$, in the main limit 
theorem, Theorem \ref{t:main.max} below. Equivalently, we will look at
the max-increments of the process $U_{\alpha,\beta}$ in
\eqref{e:represent.U}. Let 
\begin{equation}  \label{e:max.increment.2}
V_n^{(\alpha,\beta)} = \sup \bigl\{ \, j_k : n^{\beta}-1 < s_k \leq
n^{\beta} \bigr\}\,,  \ \ n=1,2,\dots\,. 
\end{equation}
An important observation is that this sequence is independent only
when $\beta=1$, that is, when $Z_{\alpha,\beta}$ and $U_{\alpha,\beta}$
coincide with the extremal $\alpha$-Fr\'echet process. In general, we
have the following statement.  

\begin{proposition} 
The stationary process $(V_n^{(\alpha,\beta)}, \, n \geq 1)$ is mixing and is generated by a dissipative flow. 
\end{proposition}
\begin{proof}
Notice that
$$
(V_n^{(\alpha,\beta)}, \, n \geq 1) \stackrel{d}{=} \bigl(
M_{\alpha}((n^{\beta}-1, n^{\beta}]), \, n \geq 1\bigr)\,, 
$$
where $M_{\alpha}$ is an independently scattered $\alpha$-Fr\'echet
sup-measure on  $\bbr_+$ with the Lebesgue control measure. See
\cite{stoev:taqqu:2005}. It follows by Theorem 3.4 in
\cite{stoev:2008}  that $(V_n^{(\alpha,\beta)}, \, n \geq 1)$ is
mixing. Further, $\sum_{n=1}^{\infty}  \one_{(n^{\beta}-1, n^{\beta}]
}(x) < \infty$  a.e., and  Theorem 5.2 of \cite{wang:stoev:2010} shows
that $V_n^{(\alpha,\beta)}$ is  generated by a dissipative flow. 
\end{proof}

The main result in Theorem \ref{t:main.max} below deals with the
partial maxima of stationary S$\alpha$S
processes for which the underlying flows are
conservative; these are associated with long memory in the
process. The resulting limiting process is not an extremal process,
hence its max-increments are dependent. The memory in the original
S$\alpha$S is long enough to guarantee that. On the other hand, the 
max-increments of the latter are generated by a dissipative flow,
which is associated with shorter memory. We observe, therefore,
another example of ``memory reduction'' by the operation of taking
partial maxima.  

We finish this section by mentioning that an immediate conclusion from
\eqref{e:represent.U} is the following representation of the process
$Z_{\alpha, \beta}$ on the interval $[0,1]$:
\begin{equation} \label{e:represent.Z}
\left( Z_{\alpha, \beta}(t), \, 0 \leq t \leq 1 \right) \stackrel{d}{=} \left( \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \one_{\{ V_j \leq t \}}, \, 0 \leq t \leq 1 \right)\,,
\end{equation}
where $\Gamma_j$, $j=1,2,\dots$, are arrival times of a unit rate
Poisson process on $(0,\infty)$, and $(V_j)$ are i.i.d. random
variables with $P(V_1 \leq x) = x^{\beta}$, $0 < x \leq 1$,
independent of $(\Gamma_j)$.  





\section{A Functional Limit Theorem for Partial Maxima} \label{sec:FLTPM}

 
In this section we state and prove our main result, a functional limit
theorem for the partial maxima of the process $\BX=(X_1,X_2,\dots)$
given in (\ref{e:underlying.proc}). Recall that  $T$ is a conservative, ergodic and
measure preserving map on a $\sigma$-finite, infinite, measure space 
$(E,\mathcal{E},\mu)$. We will assume that $T$ is a pointwise dual
ergodic map with normalizing sequence $(a_n)$ that is regularly
varying with exponent $1-\beta$; equivalently, the wandering sequence
$(w_n)$ in \eqref{e:wanderingrate} is assumed to be regularly
varying with exponent $\beta$. Crucially, we will assume that 
$1/2 < \beta < 1$. See Remark \ref{rk:other.beta} after the statement
of Theorem \ref{t:main.max} below. 

Define
\begin{equation} \label{e:b.n}
b_n = \left( \int_E \max_{1 \leq k \leq n}\bigl| f\circ
  T^n(x)\bigr|^{\alpha} \mu(dx) \right)^{1/\alpha}, \ \ \ n=1,2,\dots\,.
\end{equation}
The sequence $(b_n)$ is known to play an important role in the rate of
growth of partial maxima of an $\alpha$-stable process of the type
\eqref{e:underlying.proc}. It also turns out to be a proper
normalizing sequence for our functional limit theorem. In
\cite{samorodnitsky:2004a} it was shown that, for a canonical kernel 
(\ref{e:general.rosinski}), if the map $T$ is 
conservative, then the sequence $(b_n)$ grows at a rate strictly
slower than $n^{1/\alpha}$. The extra assumptions imposed in this
paper will guarantee a more precise statement:  $(b_n) \in
RV_{\beta/\alpha}$. Specifically, 
\begin{equation}  \label{e:RV.exp.bn}
\lim_{n\to\infty}\frac{b_n^\alpha}{ w_n}=\| f\|_\infty
\end{equation} 
(where $(w_n)$ is the wandering sequence). 
This fact has an interesting message, because it explicitly shows
that the rate of growth of the partial maxima is determined both by
the heaviness of the marginal tails (through $\alpha$) and by the
length of memory (through $\beta$). 

In contrast, if the map $T$ has a non-trivial dissipative component,
then the sequence $(b_n)$ grows at the rate  $n^{1/\alpha} $, and so
do the partial maxima of a stationary \SaS\ process; see
\cite{samorodnitsky:2004a}. This is the limiting case of the setup in
the present paper, as  $\beta$ gets closer to $1$. Intuitively, the
smaller is $\beta$, the longer is the  memory in the process. 

The functional limit theorem in Theorem
\ref{t:main.max} below 
involves weak convergence in the space $D[0,\infty)$ equipped with two
different topologies, the Skorohod $J_1$-topology and the Skorohod
$M_1$-topology, introduced in \cite{skorohod:1956}. The details could
be found, for instance, in \cite{billingsley:1999} (for the
$J_1$-topology), and in \cite{whitt:2002} (for the
$M_1$-topology). See Remark \ref{rk:topologies}. 

We recall the tail constant of an $\alpha$-stable random variable given by 
$$
C_{\alpha} = \left( \int_0^{\infty} x^{-\alpha} \sin x \,  dx
\right)^{-1} = \begin{cases} (1-\alpha) / \bigl(\Gamma(2-\alpha) \cos(\pi
  \alpha / 2)\bigr) & \ \ \text{if } \alpha \neq 1, \\ 
2 / \pi  & \ \ \text{if } \alpha=1;  
\end{cases}
$$
see \cite{samorodnitsky:taqqu:1994}. 

\begin{theorem}  \label{t:main.max}
Let $T$ be a conservative, ergodic and measure preserving map on a
$\sigma$-finite infinite measure space $(E,\mathcal{E},\mu)$. Assume
that $T$ is a pointwise dual ergodic map with normalizing 
sequence $(a_n) \in RV_{1-\beta}$, $0\leq \beta\leq 1$. Let $f\in
L^{\alpha}(\mu)\cap L^{\infty}(\mu)$, and assume that $f$ is supported by
a uniform set $A$ for $T$. Let $\alpha>0$. Then the sequence $(b_n)$
in \eqref{e:b.n} satisfies \eqref{e:RV.exp.bn}.

Assume now that $0<\alpha<2$ and $1/2<\beta<1$. If $M$ is a S$\alpha$S
random measure on $(E,\mathcal{E})$ with 
control measure $\mu$, then the stationary S$\alpha$S process $\BX$
given in (\ref{e:underlying.proc}) satisfies 
\begin{equation} \label{e:weak.conv}
\Bigl( \frac{1}{b_n} \max_{1 \leq k \leq \lfloor nt \rfloor}|X_k|, \,
t\geq 0\Bigr)  \Rightarrow \bigl( C_{\alpha}^{1/\alpha} Z_{\alpha, \beta}(t), \,
t\geq 0\bigr) \ \ \ \text{in } D[0,\infty)
\end{equation} 
in the Skorohod $M_1$-topology. Moreover, if $f = \one_A$, then the
above convergence occurs in the Skorohod $J_1$-topology as well. 
\end{theorem}

\begin{remark} \label{rk:other.beta}
{\rm
Convergence to a Fr\'echet  limit in \eqref{e:weak.conv} no longer
holds in the range $0<\beta<1/2$. We will discuss elsewhere what
non-Fr\'echet limiting processes may appear in that case. In the
boundary case $\beta=1/2$, however, the statement \eqref{e:weak.conv}
may still hold. This will be the case when the ``one jump'' property
\eqref{e:single.poisson.jump} below is satisfied. This is the case,
for example, for the Markov shift operator $T$ presented at the end
of the paper. See also \cite{samorodnitsky:2004a}. 
}
\end{remark}

\begin{remark} \label{rk:topologies}
{\rm
It is not difficult to see why the weak convergence in
\eqref{e:weak.conv} holds in the $J_1$-topology for indicator
functions, but only in the $M_1$-topology in general. Indeed, for a
general function $f$ the values of the process $M_{\lfloor
n\cdot\rfloor}(f)$ in Proposition \ref{p:max.ergodic} may have multiple
jumps on the time scale $o(n)$ before reaching the limiting value
$\|f\|_\infty$. Since the limiting process has a single jump, one does
not expect the $J_1$-convergence. On the other hand, if $f = \one_A$,
then the value $\|f\|_\infty=1$ is reached in a single jump, matching
the single jump in the limiting process.  
}
\end{remark}
\begin{proof}[Proof of Theorem \ref{t:main.max}]
We start with verifying \eqref{e:RV.exp.bn}. Obviously, 
$$
b_n^{\alpha} \leq \, \| f \|_{\infty} \mu (\varphi_A \leq n)\,,
$$
and the definition of the wandering rate sequence gives us the upper bound
$$
\limsup_{n\to\infty}\frac{b_n^\alpha}{ w_n}\leq \| f\|_\infty\,.
$$
On the other hand, take an arbitrary  $\epsilon \in \bigl( 0,\|
f \|_{\infty}\bigr)$. The set 
$$
B_{\epsilon} = \bigl\{x \in A: |f(x)| \geq \, \| f \|_{\infty} - \epsilon \bigr\}\,.
$$
is a uniform set for $T$. Let $\bigl( w_n^{(\vep)}\bigr)$ be the
corresponding wandering rate sequence. Then a  lower bound for $b_n^{\alpha}$ is obtained by 
$$
b_n^{\alpha} \geq \bigl( \| f \|_{\infty} - \epsilon \,
\bigr) \, \mu \Bigl( \bigcup_{j=1}^n T^{-j} B_{\epsilon} \Bigr)\,, 
$$
so
$$
\liminf_{n\to\infty}\frac{b_n^\alpha}{ w_n} =
\liminf_{n\to\infty}\frac{b_n^\alpha}{ w_n^{(\vep)}} \geq
\| f \|_{\infty} - \epsilon\,.
$$
Letting $\vep\to 0$, we obtain \eqref{e:RV.exp.bn}. 

Suppose now that $0<\alpha<2$ and $1/2<\beta<1$. We continue with
proving convergence in the finite dimensional distributions in
\eqref{e:weak.conv}. Since for random elements in $D[0,\infty)$ with
nondecreasing sample paths, weak convergence in the $M_1$-topology is
implied by the finite-dimensional weak convergence, this will also
establish \eqref{e:weak.conv} in the sense of weak convergence in the
$M_1$-topology. 

Fix $0 = t_0< t_1 < \dots < t_d$, $d \geq 1$. We may and will assume
that $t_d \leq 1$. We use a series representation of the random vector
$(X_1, \dots, X_n)$: with $f_k=f\circ T^k$, $k=1,2,\ldots$, 
\begin{equation}  \label{e:series.max}
(X_k, \, k=1,\dots,n) \stackrel{d}{=} \left( b_n \,
  C_{\alpha}^{1/\alpha} \sum_{j=1}^{\infty} \epsilon_j
  \Gamma_j^{-1/\alpha} \frac{f_k(U_j^{(n)})}{\max_{1 \leq i \leq n}
    |f_i(U_j^{(n)})|}, \, k=1,\dots,n \right)\,. 
\end{equation}
Here $(\epsilon_j)$ are i.i.d. Rademacher random variables
(symmetric random variables with values $\pm
1$), $(\Gamma_j)$ are the arrival times of a unit rate Poisson
process on $(0,\infty)$, and $(U_j^{(n)})$ are i.i.d. $E$-valued random
variables with the common law $\eta_n$ defined by 
\begin{equation} \label{e:eta.n}
\frac{d\eta_n}{d\mu}(x) = \frac{1}{b_n^{\alpha}} \max_{1 \leq k \leq n} |f_k(x)|^{\alpha}\,,  \ \ x \in E\,.
\end{equation}
The sequences $(\epsilon_j)$, $(\Gamma_j)$, and $(U_j^{(n)})$ are
taken to be independent. We refer to Section 3.10 of
\cite{samorodnitsky:taqqu:1994} for series representations of
$\alpha$-stable random vectors.  The representation
\eqref{e:series.max} was also used in \cite{samorodnitsky:2004a}, and
the argument below is structured similarly to the corresponding
argument {\it ibid.} 

The crucial consequence of the assumption $1/2<\beta<1$ is that, in
the series representation (\ref{e:series.max}), only the largest
Poisson jump will play an important role. It is shown in
\cite{samorodnitsky:2004a} that, under the assumptions of Theorem
\ref{t:main.max}, for every $\eta>0$, 
\begin{equation}  \label{e:single.poisson.jump}
\varphi_n(\eta) \equiv P \left( \bigcup_{k=1}^n \left\{
    \Gamma_j^{-1/\alpha} \frac{|f_k(U_j^{(n)})|}{\max_{1 \leq i \leq
        n}|f_i(U_j^{(n)})|} > \eta \ \ \text{for at least 2 different
    } j=1,2,\dots \right\} \right) \to 0 
\end{equation}
as $n \to \infty$. 

We will proceed in two steps. First, we will prove that 
\begin{equation}  \label{e:fidi.Malphabeta}
\left( \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \frac{\max_{1 \leq
      k \leq \lfloor nt_i \rfloor} |f_k(U_j^{(n)})|}{\max_{1 \leq k
      \leq n} |f_k(U_j^{(n)})|}, \, i=1,\dots,d \right) \Rightarrow
\left( Z_{\alpha, \beta}(t_i), \, i=1,\dots,d \right) \ \ \ \text{in }
\bbr_+^d\,. 
\end{equation}
Next, we will prove that, for fixed  $\lambda_1, \dots, \lambda_d >
0$, for every $0<\delta<1$, 
$$
P \bigl( b_n^{-1} \max_{1 \leq k \leq \lfloor nt_i \rfloor} |X_k| > \lambda_i, \ \ i=1,\dots,d \bigr) 
$$
\begin{equation}  \label{e:upp.bdd.max}
\leq P \left( C_{\alpha}^{1/\alpha} \bigvee_{j=1}^{\infty}
  \Gamma_j^{-1/\alpha} \frac{\max_{1 \leq k \leq \lfloor nt_i
      \rfloor}|f_k(U_j^{(n)})|}{\max_{1 \leq k \leq
      n}|f_k(U_j^{(n)})|} > \lambda_i(1-\delta), \ \ i=1,\dots,d
\right) + o(1) 
\end{equation}
and that
$$
P \bigl( b_n^{-1} \max_{1 \leq k \leq \lfloor nt_i \rfloor} |X_k| > \lambda_i, \ \ i=1,\dots,d \bigr)
$$
\begin{equation}  \label{e:lower.bdd.max}
\geq P \left( C_{\alpha}^{1/\alpha} \bigvee_{j=1}^{\infty}
  \Gamma_j^{-1/\alpha} \frac{\max_{1 \leq k \leq \lfloor nt_i
      \rfloor}|f_k(U_j^{(n)})|}{\max_{1 \leq k \leq
      n}|f_k(U_j^{(n)})|} > \lambda_i(1+\delta), \ \ i=1,\dots,d
\right) + o(1)\,. 
\end{equation}
Since the Fr\'echet distribution is continuous, the weak convergence
$$
\Bigl( b_n^{-1} \max_{1 \leq k \leq \lfloor nt_i \rfloor} |X_k|, \ \
i=1,\dots,d \Bigr) \Rightarrow
\left( Z_{\alpha, \beta}(t_i), \, i=1,\dots,d \right) \ \ \ \text{in }
\bbr_+^d
$$
will follow by taking $\delta$ arbitrarily small. 

We start with proving \eqref{e:fidi.Malphabeta}.  For $n=1,2,\ldots$, 
$N_n=\sum_{j=1}^{\infty} \delta_{(\Gamma_j, U_j^{(n)})}$  is a Poisson
random measure on $(0,\infty)\times \cup_{k=1}^{n} T^{-k}A$ 
with mean measure $\lambda \times \eta_n$. Define a map $S_n:\bbr_+
\times \cup_{k=1}^{n} T^{-k}A  \to \bbr_+^d$ by 
$$
S_n(r,x) = r^{-1/\alpha} \bigl( M_n(f)(x) \bigr)^{-1} \bigl(
M_{\lfloor nt_1 \rfloor}(f)(x), \dots, M_{\lfloor nt_d \rfloor}(f)(x)
\bigr)\,, \ \ r > 0, \ x \in \cup_{k=1}^{n} T^{-k}A\,. 
$$
Then, for $\lambda_1, \dots, \lambda_d > 0$, 
$$
P \left( \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \frac{\max_{1
      \leq k \leq \lfloor nt_i \rfloor} |f_k(U_j^{(n)})|}{\max_{1 \leq
      k \leq n} |f_k(U_j^{(n)})|} \leq \lambda_i\,, \ \ i=1,\dots,d
\right) 
$$
$$
=P \left[  N_n\left(  S_n^{-1} \Bigl( (0,\lambda_1] \times \dots \times (0,\lambda_d] \Bigr)^c\right) = 0 \right]
$$
$$
=\exp \left\{ - (\lambda \times \eta_n) \left( S_n^{-1} \Bigl(
    (0,\lambda_1] \times \dots \times (0,\lambda_d] \Bigr)^c
  \right)\right\} 
$$
$$
=\exp \left\{ - (\lambda \times \eta_n) \left\{(r,x): \bigvee_{j=1}^d
    \lambda_j^{-\alpha} \frac{\bigl(M_{\lfloor nt_j
        \rfloor}(f)(x)\bigr)^{\alpha}}{\bigl(M_n(f)(x)\bigr)^{\alpha}}
    > r \right\} \right\} 
$$
$$
= \exp \left\{ -b_n^{-\alpha} \int_E \bigvee_{j=1}^d \lambda_j^{-\alpha} M_{\lfloor nt_j \rfloor}(f)^{\alpha} d\mu \right\}\,.
$$ 
We use \eqref{e:RV.exp.bn} and the weak convergence in
Proposition \ref{p:max.ergodic} to obtain
$$
b_n^{-\alpha} \int_E \bigvee_{j=1}^d \lambda_j^{-\alpha} M_{\lfloor nt_j \rfloor}(f)^{\alpha} d\mu
\sim \| f \|_{\infty}^{-1} \int_E \bigvee_{j=1}^d \lambda_j^{-\alpha} M_{\lfloor nt_j
  \rfloor}(f)^{\alpha} d\mu_n 
$$
$$
\to \int_{\Omega^{\prime}}
\bigvee_{j=1}^d \lambda_j^{-\alpha} \one_{\{ V_{\beta} \leq t_j \}}
dP^{\prime}   = \sum_{i=1}^d (t_i^{\beta} - t_{i-1}^{\beta}) \left(
  \bigwedge_{j=i}^d \lambda_j \right)^{-\alpha}.
$$
Therefore, 
$$
P \left( \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \frac{\max_{1 \leq k \leq \lfloor nt_i \rfloor} |f_k(U_j^{(n)})|}{\max_{1 \leq k \leq n} |f_k(U_j^{(n)})|} \leq \lambda_i\,, \ \ i=1,\dots,d \right)
$$
$$
\to \exp \left\{ - \sum_{i=1}^d (t_i^{\beta} - t_{i-1}^{\beta}) \left(
    \bigwedge_{j=i}^d \lambda_j \right)^{-\alpha} \right\}
= P \bigl( Z_{\alpha, \beta}(t_i) \leq \lambda_i, \ \ i=1,\dots,d \bigr) \,.
$$
The claim (\ref{e:fidi.Malphabeta}) has, consequently, been proved. 

We continue with the statements (\ref{e:upp.bdd.max}) and (\ref{e:lower.bdd.max}). 
Since the arguments are very similar,  we only prove (\ref{e:upp.bdd.max}). 
Let $K \in \bbn$ and $0 < \epsilon < 1$ be constants so that 
$$
K+1 > \frac{4}{\alpha} \ \ \ \text{and } \ \ \delta - \epsilon K > 0\,.
$$
Then 
$$
P \bigl( b_n^{-1} \max_{1 \leq k \leq \lfloor nt_i \rfloor} |X_k| > \lambda_i, \ \ i=1,\dots,d \bigr) 
$$
$$
\leq P \left( C_{\alpha}^{1/\alpha} \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \frac{\max_{1 \leq k \leq \lfloor nt_i \rfloor}|f_k(U_j^{(n)})|}{\max_{1 \leq k \leq n}|f_k(U_j^{(n)})|} > \lambda_i(1-\delta), \ \ i=1,\dots,d \right) + \varphi_n \bigl( C_{\alpha}^{-1/\alpha} \epsilon \min_{1 \leq i \leq d} \lambda_i \bigr)
$$
$$
+ \sum_{i=1}^d P \Biggl( C_{\alpha}^{1/\alpha} \max_{1 \leq k \leq \lfloor nt_i \rfloor} \left| \sum_{j=1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \frac{f_k(U_j^{n})}{\max_{1 \leq l \leq n}|f_l(U_j^{(n)})|} \right| > \lambda_i\,, 
$$
$$
C_{\alpha}^{1/\alpha} \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \frac{\max_{1 \leq k \leq \lfloor nt_i \rfloor}|f_k(U_j^{(n)})|}{\max_{1 \leq k \leq n}|f_k(U_j^{(n)})|} \leq \lambda_i(1-\delta)\,, \ \ \text{and for each } m=1,\dots,n,
$$
$$
C_{\alpha}^{1/\alpha} \Gamma_j^{-1/\alpha} \frac{|f_m(U_j^{(n)})|}{\max_{1 \leq l \leq n}|f_l(U_j^{(n)})|} > \epsilon \min_{1 \leq l \leq d}\lambda_l \ \ \text{for at most one } j=1,2,\dots \Biggr)\,.
$$
By (\ref{e:single.poisson.jump}), it is enough to show that for all $\lambda>0$ and $0 \leq t \leq 1$, 
\begin{equation}  \label{e:neg.term}
P \Biggl( C_{\alpha}^{1/\alpha} \max_{1 \leq k \leq \lfloor nt \rfloor} \left| \sum_{j=1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \frac{f_k(U_j^{(n)})}{\max_{1 \leq i \leq n}|f_i(U_j^{(n)})|} \right| > \lambda\,, 
\end{equation}
$$
C_{\alpha}^{1/\alpha} \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \frac{\max_{1 \leq k \leq \lfloor nt \rfloor}|f_k(U_j^{(n)})|}{\max_{1 \leq k \leq n}|f_k(U_j^{(n)})|} \leq \lambda(1-\delta)\,, \ \ \text{and for each } m=1,\dots,n,
$$
$$
C_{\alpha}^{1/\alpha} \Gamma_j^{-1/\alpha} \frac{|f_m(U_j^{(n)})|}{\max_{1 \leq i \leq n}|f_i(U_j^{(n)})|} > \epsilon \lambda \ \ \text{for at most one } j=1,2,\dots \Biggr) \to 0\,.
$$
For every $k=1,2,\dots,n$, the Poisson random measure represented by the points $$
\bigl( \epsilon_j \Gamma_j^{-1/\alpha} f_k(U_j^{(n)}) \bigl( \max_{1 \leq i \leq n}|f_i(U_j^{(n)})| \bigr)^{-1}, \, j=1,2,\dots \bigr)
$$
has the same mean measure as that represented by the points 
$$
\bigl( \epsilon_j \Gamma_j^{-1/\alpha} \| f \|_{\alpha} b_n^{-1}, \, j=1,2,\dots \bigr)\,.
$$
Hence, these  two Poisson random measures coincide
distributionally. We conclude that the probability in  (\ref{e:neg.term}) is bounded by 
$$
\sum_{k=1}^{\lfloor nt \rfloor} P \Biggl( C_{\alpha}^{1/\alpha} \left| \sum_{j=1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \frac{f_k(U_j^{(n)})}{\max_{1 \leq i \leq n}|f_i(U_j^{(n)})|}  \right| > \lambda\,, \ \ C_{\alpha}^{1/\alpha} \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \frac{f_k(U_j^{(n)})}{\max_{1 \leq i \leq n}|f_i(U_j^{(n)})|} \leq \lambda(1-\delta)\,,
$$
$$
C_{\alpha}^{1/\alpha} \Gamma_j^{-1/\alpha} \frac{|f_k(U_j^{(n)})|}{\max_{1 \leq i \leq n}|f_i(U_j^{(n)})|} > \epsilon \lambda \ \ \text{for at most one } j=1,2,\dots \Biggr)
$$
$$
= \lfloor nt \rfloor P \Biggl( C_{\alpha}^{1/\alpha} \left| \sum_{j=1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \right| > \lambda \| f \|_{\alpha}^{-1} b_n\,, \ \ C_{\alpha}^{1/\alpha} \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha} \leq \lambda (1-\delta) \| f \|_{\alpha}^{-1} b_n\,,
$$
$$
C_{\alpha}^{1/\alpha} \Gamma_j^{-1/\alpha} > \epsilon \lambda \| f \|_{\alpha}^{-1} b_n \ \ \text{for at most one } j=1,2,\dots \Biggr) 
$$
$$
\leq n P \left( C_{\alpha}^{1/\alpha} \left| \sum_{j=K+1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \right| > (\delta - \epsilon K) \lambda \| f \|_{\alpha}^{-1} b_n \right)
$$
$$
\leq \frac{n \| f \|_{\alpha}^4 C_{\alpha}^{4/\alpha} }{(\delta - \epsilon K)^4 \lambda^4 b_n^4} \ E \left| \sum_{j=K+1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \right|^4\,.
$$
Due to the choice $K+1>4/\alpha$, 
$$
E \left| \sum_{j=K+1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \right|^4 < \infty;
$$
see \cite{samorodnitsky:2004a} for a detailed proof. Since $n/b_n^4
\to 0$ as $n \to \infty$, (\ref{e:neg.term}) follows. 

Suppose now that $f = \one_A$. In that case the probability measure
$\eta_n$ defined in \eqref{e:eta.n} coincides with the probability
measure $\mu_n$ of Proposition \ref{p:max.ergodic}.  
In order to prove weak convergence in
the $J_1$-topology, we will use a truncation argument. First of all,
we may and will restrict ourselves to the space $D[0,1]$. Let
$K=1,2,\ldots$. First of all, we show, in the 
notation of \eqref{e:represent.Z}, the convergence 
\begin{equation}  \label{e:fidi.truncate}
\left( C_{\alpha}^{1/\alpha} \max_{1 \leq k \leq \lfloor nt \rfloor}
  \left| \sum_{j=1}^K \epsilon_j \Gamma_j^{-1/\alpha} \one_A \circ
    T^k(U_j^{(n)}) \right|, \ 0\leq t\leq 1  \right) \Rightarrow
\left( C_{\alpha}^{1/\alpha} \bigvee_{j=1}^K \Gamma_j^{-1/\alpha}
  \one_{\{ V_j \leq t \}}, \ 0\leq t\leq 1 \right) 
\end{equation}
in the $J_1$-topology on $D[0,1]$. Indeed, by
\eqref{e:single.poisson.jump}, outside of an event of asymptotically
vanishing probability, the process in the left hand side of
\eqref{e:fidi.truncate}  is 
\begin{equation}  \label{e:fidi.truncate1}
\left( C_{\alpha}^{1/\alpha} \bigvee_{j=1}^K \Gamma_j^{-1/\alpha}
  \max_{1 \leq k \leq \lfloor nt \rfloor} \one_A \circ
  T^k(U_j^{(n)}), \ 0\leq t\leq 1 \right) \,.
\end{equation}
By Proposition \ref{p:max.ergodic}, we can put all the random
variables involved on the same probability space so that the time of
the single step in the $j$th term in \eqref{e:fidi.truncate1}
converges a.s. for each $j=1,\ldots, K$ to $V_j$. Then, trivially, the
process in \eqref{e:fidi.truncate1} converges a.s. in the
$J_1$-topology on $D[0,1]$ to the  process in the right hand side 
of \eqref{e:fidi.truncate}. Therefore, the weak convergence in
\eqref{e:fidi.truncate} follows. 

Next, we note that in the $J_1$-topology on the space $D[0,1]$, 
$$
\left( C_{\alpha}^{1/\alpha} \bigvee_{j=1}^K \Gamma_j^{-1/\alpha}
  \one_{\{ V_j \leq t \}}, \ 0\leq t\leq 1\right) \to \left(
  C_{\alpha}^{1/\alpha} \bigvee_{j=1}^{\infty} \Gamma_j^{-1/\alpha}
  \one_{\{ V_j \leq t_i \}} \ 0\leq t\leq 1\right)   \ \ \ \text{as } K \to \infty \ \ \ \text{a.s.}
$$
This is so  because, as $K \to \infty$,
$$
\sup_{0 \leq t \leq 1} \left( \bigvee_{j=1}^{\infty}
  \Gamma_j^{-1/\alpha} \one_{\{ V_j \leq t \}} - \bigvee_{j=1}^K
  \Gamma_j^{-1/\alpha} \one_{\{ V_j \leq t \}} \right) 
 \leq \Gamma_{K+1}^{-1/\alpha} \to 0 \ \ \ \text{a.s.}.
$$

According to Theorem 3.2 in \cite{billingsley:1999}, the
$J_1$-convergence in \eqref{e:weak.conv} will follow once we show that 
$$
\lim_{K \to \infty} \limsup_{n \to \infty} P \left( \max_{1 \leq k
    \leq n} \left| \sum_{j=K+1}^{\infty} \epsilon_j
    \Gamma_j^{-1/\alpha} \one_A \circ T^k(U_j^{(n)}) \right| >
  \epsilon \right) = 0 
$$
for every $\epsilon>0$. Write 
$$
P \left( \max_{1 \leq k \leq n} \left| \sum_{j=K+1}^{\infty}
    \epsilon_j \Gamma_j^{-1/\alpha} \one_A \circ T^k(U_j^{(n)})
  \right| > \epsilon \right)
$$
$$
\leq \int_0^{(\epsilon/2)^{-\alpha}} e^{-x} \frac{x^{K-1}}{(K-1)!} dx
+ \int_{(\epsilon/2)^{-\alpha}}^{\infty} e^{-x} \frac{x^{K-1}}{(K-1)!}
\, P \left( \max_{1 \leq k \leq n} \left| \sum_{j=1}^{\infty}
    \epsilon_j \bigl( \Gamma_j + x \bigr)^{-1/\alpha} \one_A \circ
    T^k(U_j^{(n)}) \right| > \epsilon \right) dx \,.
$$
Clearly,  the first term vanishes when $K \to \infty$. Therefore, it
is sufficient to show that for every $x \geq (\epsilon/2)^{-\alpha}$,
\begin{equation}  \label{e:upper.app.Gamma1}
P \left( \max_{1 \leq k \leq n} \left| \sum_{j=1}^{\infty} \epsilon_j
    \bigl( \Gamma_j + x \bigr)^{-1/\alpha} \one_A \circ T^k(U_j^{(n)})
  \right| > \epsilon \right) \to 0 
\end{equation}
as $n \to \infty$. \\
To  this end, choose $L \in \bbn$ and $0 < \xi < 1/2$ so that
\begin{equation}  \label{e:const.rest2}
L+1 > \frac{4}{\alpha} \ \ \ \text{and }  \ \ \frac{1}{2} - \xi L > 0\,.
\end{equation}
By \eqref{e:single.poisson.jump} we can write 
\begin{equation}  \label{e:one.term.vanish}
P \left( \max_{1 \leq k \leq n} \left| \sum_{j=1}^{\infty} \epsilon_j \bigl( \Gamma_j + x \bigr)^{-1/\alpha} \one_A \circ T^k(U_j^{(n)}) \right| > \epsilon \right) 
\end{equation}
$$
\leq P \Biggl( \max_{1 \leq k \leq n} \left| \sum_{j=1}^{\infty}
  \epsilon_j \bigl( \Gamma_j + x \bigr)^{-1/\alpha} \one_A \circ
  T^k(U_j^{(n)}) \right| > \epsilon\,,   \ \ \text{and for each } m = 1,\dots, n,
$$
$$
\bigl( \Gamma_j + x \bigr)^{-1/\alpha} \one_A \circ T^m(U_j^{(n)}) > \xi \epsilon \ \ \text{for at most one } j =1,2,\dots \Biggr) + o(1)\,.
$$
Notice that for every $k=1,\dots, n$, the Poisson random measure represented by the points 
$$
\bigl( \epsilon_j \bigl( \Gamma_j + x \bigr)^{-1/\alpha} \one_A \circ T^k(U_j^{(n)}), \, j=1,2,\dots \bigr)
$$
is distributionally equal to the Poisson random measure represented by the points 
$$
\bigl( \epsilon_j \bigl( b_n^{\alpha} \mu(A)^{-1} \Gamma_j +x \bigr)^{-1/\alpha}, \, j=1,2,\dots \bigr)\,.
$$
Therefore, the first term on the right hand side of (\ref{e:one.term.vanish}) can be bounded by 
$$
\sum_{k=1}^n P \Biggl( \left| \sum_{j=1}^{\infty} \epsilon_j \bigl(
  \Gamma_j + x \bigr)^{-1/\alpha} \one_A \circ T^k(U_j^{(n)}) \right|
> \epsilon\,,  
$$
$$
\bigl( \Gamma_j + x \bigr)^{-1/\alpha} \one_A \circ T^k(U_j^{(n)}) > \xi \epsilon \ \ \text{for at most one } j=1,2,\dots \Biggr)
$$
$$
= nP \Biggl( \left| \sum_{j=1}^{\infty} \epsilon_j \bigl( b_n^{\alpha}
  \mu(A)^{-1} \Gamma_j + x \bigr)^{-1/\alpha} \right| > \epsilon\,,  
$$
$$
\bigl( b_n^{\alpha} \mu(A)^{-1} \Gamma_j + x \bigr)^{-1/\alpha} > \xi \epsilon \ \ \text{for at most one } j = 1,2,\dots \Biggr)
$$
$$
\leq n P \left( \left| \sum_{j=L+1}^{\infty} \epsilon_j \bigl( b_n^{\alpha} \mu(A)^{-1} \Gamma_j + x \bigr)^{-1/\alpha} \right| > \left( \frac{1}{2} - \xi L \right)\epsilon \right)\,.
$$
In the last step we used the fact that, for $x \geq
(\epsilon/2)^{-\alpha}$, the magnitude of each term in the infinite
sum does not exceed $\epsilon/2$. By the contraction inequality for
Rademacher series (see e.g. Proposition 1.2.1 of
\cite{kwapien:woyczynski:1992}),  
$$
nP \left( \left| \sum_{j=L+1}^{\infty} \epsilon_j \bigl( b_n^{\alpha} \mu(A)^{-1} \Gamma_j + x \bigr)^{-1/\alpha} \right| > \left( \frac{1}{2} - \xi L \right)\epsilon \right) 
$$
$$
\leq 2 n P \left( \left| \sum_{j=L+1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \right| > \left( \frac{1}{2} - \xi L \right)\epsilon \mu(A)^{-1/\alpha} b_n \right)\,.
$$
As before, by Markov's inequality and using the 
constraints of the constants $L \in \bbn$ and $0 < \xi < 1/2$ given in (\ref{e:const.rest2}), 
$$
2n P \left( \left| \sum_{j=L+1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \right| > \left( \frac{1}{2} - \xi L \right)\epsilon \mu(A)^{-1/\alpha} b_n \right) \leq \frac{2n\mu(A)^{4/\alpha}}{(2^{-1}-\xi L)^4 \epsilon^4 b_n^4} \ E \left| \sum_{j=L+1}^{\infty} \epsilon_j \Gamma_j^{-1/\alpha} \right|^4 \to 0
$$
as $n \to \infty$ and, hence, (\ref{e:upper.app.Gamma1}) follows. 
\end{proof}


\begin{remark}
{\rm
There is not doubt that the convergence result in Theorem
\ref{t:main.max} can be extended to more general infinitely divisible
random measures $M$ in \eqref{e:underlying.proc}, under appropriate
assumptions of regular variation of the L\'evy measure of $M$ and
integrability of the function $f$. In particular, processes
$Z_{\alpha, \beta}$ with any $\alpha>0$ are likely to appear in the
limit in \eqref{e:weak.conv} (and not only with $0<\alpha<2$ allowed
by the assumption of stability). Furthermore, the symmetry of the
process $\BX$ has very little to do with the limiting distribution of
the partial maxima. For example, a straightforward symmetrization
argument allows one to extend \eqref{e:weak.conv} to skewed
$\alpha$-stable processes, at least in the sense of convergence of
finite-dimensional distributions. The reason we decided to restrict
the presentation to the symmetric stable case had to do with a
particularly simple form of the series representation
\eqref{e:series.max} available in this case. This has allowed us to avoid
certain technicalities that might have otherwise blurred the main
message, which 
is the effect of memory on the functional limit theorem for the
partial maxima. 
}
\end{remark}

We conclude by mentioning that the result of Theorem
\ref{t:main.max} applies, among others, to the two examples in 
\cite{owada:samorodnitsky:2012}, that of a flow generated by a null
recurrent Markov chain, and that of an AFN-system. We only remind the
reader of the setup of the former example, since it also appears in
\cite{samorodnitsky:2004a}. 

Consider an irreducible null recurrent Markov chain $(x_n, \, n \geq
0)$ defined on an infinite countable state space $\bbs$ with the
transition matrix $(p_{ij})$. Let $(\pi_i, \, i \in \bbs)$ be its
unique (up to constant multiplication) invariant measure with
$\pi_0=1$. Note that $(\pi_i)$ is necessarily an infinite
measure. Define a $\sigma$-finite and infinite measure on
$(E,\mathcal{E}) = (\bbs^{\bbn}, \mathcal{B}(\bbs^{\bbn}))$ by  
$$
\mu(B) = \sum_{i \in \bbs} \pi_i P_i(B), \ \ B \subseteq \bbs^{\bbn}\,,
$$
where $P_i(\cdot)$ denotes the probability law of $(x_n)$ starting in state $i \in \bbs$. Let 
$$
T(x_0, x_1, \dots) = (x_1,x_2,\dots)
$$
be the usual left shift operator on $\bbs^{\bbn}$. Then $T$ preserves
$\mu$. Since the Markov chain is irreducible and null recurrent, $T$
is conservative and ergodic (see \cite{harris:robbins:1953}).

Let $A = \{ x \in \bbs^{\bbn}: x_0=i_0 \}$ for a fixed state $i_0\in
\bbs$, and 
$$
\varphi_A(x) = \min \{ n \geq 1: x_n \in A \}\,, \ \ x \in \bbs^{\bbn}
$$
be the first entrance time. 
Assume that
$$
\sum_{k=1}^n P_0(\varphi_A \geq k) \in RV_{\beta}
$$
for some $\beta \in (1/2, 1)$. Then all of the assumptions of Theorem
\ref{t:main.max} are satisfied for any $f\in
L^{\alpha}(\mu)\cap L^{\infty}(\mu)$,   supported by $A$; 
see \cite{aaronson:1997}. 




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\end{document}


1307.2356.extracted_bib
\begin{thebibliography}{43}
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\bibitem[\protect\citeauthoryear{Dwass}{Dwass}{1966}]{dwass:1966}
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\bibitem[\protect\citeauthoryear{Lamperti}{Lamperti}{1964}]{lamperti:1964}
---\hspace{-.1pt}---\hspace{-.1pt}--- (1964): \enquote{On extreme order
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  negative drift random walk with dependent heavy--tailed steps,} \emph{Annals
  of Applied Probability}, 10, 1025--1064.

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  Annals of Statistics}, 28, 1427--1451.

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\end{thebibliography}