# Empirical process
sampled along a stationary process

Guy Cohen and Jean-Pierre Conze  Guy Cohen, School of Electrical Engineering,
Ben-Gurion University, Israel<EMAIL_ADDRESS>Jean-Pierre Conze, IRMAR,
CNRS UMR 6625, University of Rennes, Campus de Beaulieu, 35042 Rennes Cedex,
France<EMAIL_ADDRESS>
###### Abstract.

Let $(X_{{\underline{\ell}}})_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}$ be a
real random field (r.f.) indexed by ${\mathbb{Z}}^{d}$ with common probability
distribution function $F$. Let $(z_{k})_{k=0}^{\infty}$ be a sequence in
${\mathbb{Z}}^{d}$. The empirical process obtained by sampling the random
field along $(z_{k})$ is $\sum_{k=0}^{n-1}[{\bf 1}_{X_{z_{k}}\leq s}-F(s)]$.

We give conditions on $(z_{k})$ implying the Glivenko-Cantelli theorem for the
empirical process sampled along $(z_{k})$ in different cases (independent,
associated or weakly correlated random variables). We consider also the
functional central limit theorem when the $X_{\underline{\ell}}$’s are i.i.d.

These conditions are examined when $(z_{k})$ is provided by an auxiliary
stationary process in the framework of “random ergodic theorems”.

###### Key words and phrases:

Empirical process, sampling along a stationary process, local times, Glivenko-
Cantelli theorem, functional central limit theorem, random walks

###### 2010 Mathematics Subject Classification:

Primary: 60F05, 28D05, 22D40, 60G50; Secondary: 47B15, 37A25, 37A30

###### Contents

  1. 1 General results on the empirical process along a sub-sequence
    1. 1.1 Preliminaries
    2. 1.2 A Glivenko-Cantelli type theorem
    3. 1.3 A sufficient condition for a FCLT for the sampled empirical process
  2. 2 Local times for ergodic sums
    1. 2.1 Auxiliary general results
    2. 2.2 Non centered cocycles
    3. 2.3 Counterexamples
  3. 3 Examples
    1. 3.1 Random walks
    2. 3.2 Extensions of the r.w. case
    3. 3.3 Step functions over rotations
  4. 4 About limit theorems along ergodic sums
    1. 4.1 Glivenko-Cantelli theorem along ergodic sums
    2. 4.2 Discussion: universal sequences

## Introduction

For a sequence $(X_{k})$ of real i.i.d. random variables with common
probability distribution function $F$, the empirical process is defined by
$\sum_{k=0}^{n-1}\,[{\bf 1}_{X_{k}\leq s}-F(s)]$. Recall two classical
results. (A) the Glivenko-Cantelli theorem: a.s. the sequence of empirical
distribution functions $F_{n}(s):=\frac{1}{n}\sum_{k=0}^{n-1}{\bf
1}_{X_{k}\leq s}$ converges uniformly to $F$, i.e. $\sup_{s}|F_{n}(s)-F(s)|\to
0$;  (B) a functional central limit theorem (FCLT): if the r.v.s $X_{k}$ have
a common distribution $F$ over $[0,1]$, then the process
${1\over\sqrt{n}}\sum_{k=0}^{n-1}\,[{\bf 1}_{X_{k}\leq s}-F(s)]$ converges
weakly to a Brownian bridge in the space of cadlag functions on $[0,1]$.

In this paper we study the extension of these results when the process is
sampled along a subsequence, analogously to what is done for limit theorems in
random scenery.

In the sequel, for $d\geq 1$,
$(X_{{\underline{\ell}}})_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}$ will be a
real random field (r.f.) indexed by ${\mathbb{Z}}^{d}$ defined on a
probability space $(\Omega,\mathcal{F},{\mathbb{P}})$ with common probability
distribution function $F$. The expectation on $(\Omega,{\mathbb{P}})$ is
denoted by ${\mathbb{E}}$. We consider in particular the case of a r.f. of
i.i.d. r.v.’s or of stationary associated r.v.’s.

Let $(z_{k})_{k=0}^{\infty}$ be a sequence in ${\mathbb{Z}}^{d}$. The process
obtained by sampling the random field along $(z_{k})$ is
$W_{n}(s):=\sum_{k=0}^{n-1}[{\bf 1}_{X_{z_{k}}\leq s}-F(s)]$.

We will call $W_{n}(s)$ “empirical process sampled along $(z_{k})$”, or simply
“sampled empirical process”. A general question is whether the above results
(A), (B) extend to the sampled empirical process $W_{n}(s)$, in particular
when $(z_{k})$ is given by another stationary process with values in
${\mathbb{Z}}^{d}$.

In Section 1, we give conditions on $(z_{k})$ implying that (A) and (B) are
still valid for an empirical process sampled along $(z_{k})$ in different
cases: independent, associated or weakly correlated random variables. The
conditions are expressed in terms of the following quantities associated to
the sequence $(z_{k})$ in ${\mathbb{Z}}^{d}$: local time, maximal local time
and number of self-intersections (up to time $n$) defined, for $n\geq 1$, by

$\displaystyle N_{n}({\underline{\ell}}):=\\#\\{0\leq k\leq n-1:\
z_{k}={\underline{\ell}}\\},$ (1) $\displaystyle\
M_{n}:=\max_{{\underline{\ell}}}N_{n}({\underline{\ell}}),\ V_{n}:=\\#\\{0\leq
j,k\leq n-1:\,z_{j}=z_{k}\\}.$

They satisfy $\sum_{\underline{\ell}}N_{n}({\underline{\ell}})=n$ and $n\leq
V_{n}=\sum_{\underline{\ell}}N_{n}^{2}({\underline{\ell}})\leq nM_{n}\leq
n^{2}$.

In the other sections, $(z_{k})$ is given by a stationary process (or
equivalently by the sequence $(S_{k}f(x))_{k\geq 1}$ of ergodic sums of a
function $f$ over a dynamical system).

The conditions found in Section 1 lead to study the local times, maximum
number of visits, number of self-intersections for the sequence $(S_{k}f(x))$.
General remarks are presented in Section 2. Then in Section 3, we consider two
families of examples: random walks and some ergodic sums over a rotation.

The Glivenko-Cantelli theorem along ergodic sums (extension of (A)) is
strongly related to random ergodic theorems, in particular to results in [23]
and [25]. This is discussed in the last Section 4.

Finally let us mention the quenched FCLT for the 2-parameters process

$W_{n}(s,t):=\sum_{k=0}^{{[nt]}-1}\,[{\bf 1}_{X_{Z_{k}(x)}\leq
s}-F(s)],\,(s,t)\in[0,1]^{2}.$

When $(X_{\underline{\ell}})$ is a r.f. of i.i.d. r.v.’s indexed by
${\mathbb{Z}}^{2}$ and when the sampling is provided by a 2-dimension centered
random walk $(Z_{k})$ with a moment of order 2, the weak convergence for a.e.
$x$ toward a Kiefer-Müller process can be shown. This will be the content of a
forthcoming paper.

Acknowledgements. Part of this research was done during visits of the first
author to the IRMAR at the University of Rennes 1 and of the second author to
the Center for Advanced Studies in Mathematics at Ben Gurion University. The
authors are grateful to their hosts for their support.

##  1\. General results on the empirical process along a sub-sequence

###  1.1. Preliminaries

In this subsection, results on the empirical process along a sub-sequence are
shown for independent variables, as well for some of them for wider classes
(associated, PDQ and weakly correlated random variables). We start by
recalling some notions and auxiliary results.

1) Associated variables

Definition  (cf. [17]): A finite set of real random variables ${\bf
T}=(T_{1},T_{2},\ldots,T_{n})$ is said to be associated if ${\rm Cov}[f({\bf
T}),g({\bf T})]\geq 0$, for every coordinate-wise non-decreasing functions
$f=f(x_{1},...,x_{n})$ and $g=g(x_{1},...,x_{n})$ for which
${\mathbb{E}}[f({\bf T})]$, ${\mathbb{E}}[g({\bf T})]$, ${\mathbb{E}}[f({\bf
T})\,g({\bf T})]$ exist. An infinite set of random variables is associated if
any finite subset of it is associated.

Association of random variables is preserved under taking subsets and forming
unions of independent sets (of associated random variables). In particular a
family of independent variables is associated.

Clearly, orthogonal associated random variables are independent. Examples of
(non independent) stationary associated processes with absolutely summable
series of correlations are provided by some Ising models. References to such
examples of stationary ${\mathbb{Z}}^{d}$ random fields which satisfies the
FKG inequalities and with absolutely summable correlations can be found in
Newman’s paper [26]. Notice that the FKG inequalities expresses the
association property of the r.v.’s.

2) PQD variables

Two r.v.’s $X,Y$ are called (cf. [24]) positively quadrant dependent (PQD) if,

${\mathbb{P}}(X>x,Y>y)\geq{\mathbb{P}}(X>x)\,{\mathbb{P}}(Y>y),\forall
x,y\in{\mathbb{R}}.$

The property is preserved by centering. Any pairwise associated r.v.’s are
pairwise PQD. Pairwise independent random variables are pairwise PQD
associated.

Two random variables $X$ and $Y$ are PQD if and only if for every non-
decreasing functions $f$ and $g$, $\text{Cov}(f(X),g(Y))\geq 0$ (whenever the
covariance exists) ([17, Theorem 4.4]).

3) We will use the following results:

a) Maximal inequality of Newman and Wright [27, Inequality (12)]:

If $(W_{i})$ is a sequence of centered associated, square integrable random
variables, it holds:

(2) $\displaystyle{\mathbb{P}}(\max_{1\leq j\leq
n}|\sum_{i=1}^{j}W_{i}|\geq\lambda\,\|\sum_{i=1}^{n}W_{i}\|_{2})\leq
2{\mathbb{P}}(|\sum_{i=1}^{n}W_{i}|\geq(\lambda-\sqrt{2})\,\|\sum_{i=1}^{n}W_{i}\|_{2}),\forall\lambda\geq
0.$

b) Hoeffding’s identity (see [2, Theorem 3.1])

Let $X,Y$ be random variables with finite second moments. For any absolutely
continuous functions $f,g$ on $\mathbb{R}$, such that
${\mathbb{E}}[f^{2}(X)+g^{2}(Y)]<\infty$, it holds

$\text{Cov}(f(X),g(Y))=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f^{\prime}(x)g^{\prime}(y)[{\mathbb{P}}(X>x,Y>y)-{\mathbb{P}}(X>x){\mathbb{P}}(Y>y)]dxdy.$

In particular, if $X,Y$ are PDQ random variables, if
$|f^{\prime}|,|g^{\prime}|\leq M$ a.e., we have

$|\text{Cov}(f(X),g(Y))|\leq M^{2}\text{Cov}(X,Y).$

4) Uniformity in the analogues of Glivenko-Cantelli theorem will follow from
the lemma:

######  Lemma 1.1.

[7, Lemma, p 140] Let $F_{n}$, $F$ be a family of right continuous
distributions on $\mathbb{R}$. Assume that, for each point $x$ in a dense
countable set $Q\subset\mathbb{R}$, we have $F_{n}(x)\to F(x)$. Let $J$ be the
set of jumps of $F$ and assume that $F_{n}(x)-F_{n}(x^{-})\to F(x)-F(x^{-})$
for every $x\in J$. Then $F_{n}(x)\to F(x)$ uniformly in $\mathbb{R}$.

A strong law of large numbers

First we state a law of large numbers for bounded r.v.’s valid under weak
hypotheses.

Let $(U_{\underline{\ell}})_{{\underline{\ell}}\in\mathbb{Z}^{d}}$ be a r.f.
indexed by ${\mathbb{Z}}^{d}$ of square integrable r.v’s on a probability
space $(\Omega,\mathcal{F},{\mathbb{P}})$. Let $(z_{k})_{k\geq 0}$ be a
sequence in ${\mathbb{Z}}^{d}$, $d\geq 1$, with numbers of self-intersections
$V_{n},n\geq 1$. The partial sums along $(z_{k})$ are denoted by
$\displaystyle S_{n}:=\sum_{k=0}^{n-1}U_{z_{k}}$.

By the Cauchy-Schwarz inequality, if
$\sum_{\underline{\ell}}\sup_{\underline{r}}|\langle
U_{{\underline{r}}+{\underline{\ell}}},U_{{\underline{r}}}\rangle|<+\infty$,
if holds for a finite constant $C_{0}$:

(3)
$\displaystyle\|\sum_{i=0}^{n-1}U_{z_{i}}\|_{2}^{2}=\sum_{\underline{\ell}}\sum_{\underline{r}}N_{n}({\underline{r}}+{\underline{\ell}})N_{n}({\underline{r}})\langle
U_{{\underline{r}}+{\underline{\ell}}},U_{{\underline{r}}}\rangle\leq
V_{n}\sum_{\underline{\ell}}\sup_{\underline{r}}|\langle
U_{{\underline{r}}+{\underline{\ell}}},U_{{\underline{r}}}\rangle|=C_{0}V_{n}.$

In particular if the r.f. is stationary and the series of correlations is
absolutely summable (i.e.,
$\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}|\langle
X_{\underline{0}},X_{\underline{\ell}}\rangle|<+\infty$), then the spectral
density of the r.f. exists and is the continuous non-negative function $\rho$
on ${\mathbb{T}}^{d}$ with Fourier coefficients
$\displaystyle\int_{{\mathbb{T}}^{d}}e^{2\pi
i\langle{\underline{\ell}},{\underline{t}}\rangle}\,\rho({\underline{t}})\,d{\underline{t}}=\langle
X_{\underline{0}},X_{\underline{\ell}}\rangle$ and it holds:

(4) $\displaystyle\|S_{n}\|_{2}^{2}=\|\sum_{i=0}^{n-1}U_{z_{i}}\|_{2}^{2}\leq
V_{n}\sum_{\underline{\ell}}|\langle
U_{\underline{\ell}},U_{\underline{0}}\rangle|.$

######  Proposition 1.2.

Suppose the r.v.’s $U_{\underline{\ell}}$ on $(\Omega,{\mathbb{P}})$ centered
and uniformly bounded by the same constant $K$,
$\|U_{\underline{\ell}}\|_{\infty}\leq K,\forall{\underline{\ell}}$. Assume
that $(z_{k})$ is such that

(5) $\displaystyle V_{n}\leq C_{1}{n^{2}\over(\log n)^{\beta}},\text{ for
constants }C_{1},\beta.$

1) Then, if $\beta>1$ and
$\displaystyle\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\sup_{r\in{\mathbb{Z}}^{d}}|\langle
U_{{\underline{r}}+{\underline{\ell}}},U_{{\underline{r}}}\rangle|<+\infty$,
2) or if $\beta>\zeta$ for some $\zeta\in[1,2]$ and the r.f.
$(U_{\underline{\ell}})$ is stationary with
$\displaystyle\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}|\langle
U_{\underline{\ell}},U_{{\underline{0}}}\rangle|^{\zeta}<\infty$, the (strong)
LLN holds: $\displaystyle{S_{n}(\omega)\over n}\to 0$, for ${\mathbb{P}}$-a.e
$\omega$.

Proof. 1) For convenience, if $t$ is in ${\mathbb{R}}^{+}$, we define $S_{t}$
as $S_{[t]}$. From (3) it follows

$\int({|S_{n}|\over n})^{2}\,d{\mathbb{P}}\leq C_{0}{V_{n}\over n^{2}}\leq
C_{0}C_{1}{1\over(\log n)^{\beta}}.$

Therefore, putting $\beta=1+\eta$ and $\alpha=1-\eta/2$ (which implies
$\alpha\beta>1$) we have

$\displaystyle\sum_{k}\int({|S_{2^{k^{\alpha}}}|\over
2^{{k^{\alpha}}}})^{2}\,d{\mathbb{P}}\leq
C_{0}C_{1}\sum_{k}{1\over(\log{2^{k^{\alpha}}})^{\beta}}=C^{\prime}\sum_{k}{1\over
k^{\alpha\beta}}<+\infty;$

hence: $\displaystyle\lim_{k\to+\infty}{S_{2^{k^{\alpha}}}\over
2^{{k^{\alpha}}}}=0$, a.e.

For $n\geq 1$, let $k_{n}$ be such that $2^{{(k_{n})^{\alpha}}}\leq
n<2^{{(k_{n}+1)^{\alpha}}}$ (that is: $k_{n}=[(\log_{2}n)^{1/\alpha}]$).

We put $q_{n}:=2^{{(k_{n}+1)^{\alpha}}}-2^{{(k_{n})^{\alpha}}}$ and
$p_{n}=n-2^{{(k_{n})^{\alpha}}}\leq q_{n}$.

For $q_{n}$, the following estimate holds: $\displaystyle
q_{n}=2^{{(k_{n})^{\alpha}}}(2^{{(k_{n}+1)^{\alpha}}-{(k_{n})^{\alpha}}}-1)\sim
C^{\prime\prime}{2^{{(k_{n})^{\alpha}}}\over{(k_{n})^{1-\alpha}}}$.

Using the uniform boundedness of the r.v.’s, we can write:

$\displaystyle|{S_{n}\over n}-{S_{2^{{(k_{n})^{\alpha}}}}\over
2^{{(k_{n})^{\alpha}}}}|=|{S_{2^{{(k_{n})^{\alpha}}}}+\sum_{i=2^{{(k_{n})^{\alpha}}}}^{2^{{(k_{n})^{\alpha}}}+p_{n}}U_{z_{i}}\over
2^{{(k_{n})^{\alpha}}}+p_{n}}-{S_{2^{{(k_{n})^{\alpha}}}}\over
2^{{(k_{n})^{\alpha}}}}|=|{2^{{(k_{n})^{\alpha}}}\,\sum_{i=2^{{(k_{n})^{\alpha}}}}^{2^{{(k_{n})^{\alpha}}}+p_{n}}U_{z_{i}}-p_{n}S_{2^{{(k_{n})^{\alpha}}}}\over
2^{{(k_{n})^{\alpha}}}(2^{{(k_{n})^{\alpha}}}+p_{n})}|$
$\displaystyle\leq{2^{{(k_{n})^{\alpha}}}\,\sum_{i=2^{{(k_{n})^{\alpha}}}}^{2^{{(k_{n})^{\alpha}}}+p_{n}}|U_{z_{i}}|+p_{n}|S_{2^{{(k_{n})^{\alpha}}}}|\over
2^{{(k_{n})^{\alpha}}}(2^{{(k_{n})^{\alpha}}}+p_{n})}\leq{2^{{(k_{n})^{\alpha}}}\,\sum_{i=2^{{(k_{n})^{\alpha}}}}^{2^{{(k_{n})^{\alpha}}}+q_{n}}|U_{z_{i}}|+q_{n}|S_{2^{{(k_{n})^{\alpha}}}}|\over
2^{{(k_{n})^{\alpha}}}(2^{{(k_{n})^{\alpha}}})}$
$\displaystyle\leq{q_{n}K2^{{(k_{n})^{\alpha}}}+q_{n}|S_{2^{{(k_{n})^{\alpha}}}}|\over
2^{{(k_{n})^{\alpha}}}(2^{{(k_{n})^{\alpha}}})}={q_{n}\over
2^{{(k_{n})^{\alpha}}}}(K+\,{|S_{2^{{(k_{n})^{\alpha}}}}|\over
2^{{(k_{n})^{\alpha}}}}).\leq{C\over
2{(k_{n})^{1-\alpha}}}(K+\,{|S_{2^{{(k_{n})^{\alpha}}}}|\over
2^{{(k_{n})^{\alpha}}}})\to 0.$

2) We consider now the stationary case. Since $\zeta=1$ is special case of 1),
we assume $\zeta\in]1,2]$. We put $\beta=\zeta+\eta$, where $\eta$ is $>0$ in
view of the hypothesis.

First, suppose that $\zeta=2$. Then under the hypothesis, the r.f. has a
spectral measure $\nu_{\varphi}$ absolutely continuous with respect to the
Lebesgue measure $\lambda$ on the torus with a density $\rho\in
L^{2}(d{\underline{t}})$ given by the Fourier series
$\rho({\underline{t}})=\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\langle
U_{\underline{\ell}},U_{{\underline{0}}}\rangle\,{\rm
e}^{2i\pi\langle{\underline{\ell}},{\underline{t}}\rangle}$.

Using the inequality $\lambda\\{\rho>M_{n}\\}\leq
M_{n}^{-2}\,\|\rho\|_{2}^{2}$, we can write:

$\displaystyle{\|S_{n}\|_{2}^{2}\over
n^{2}}=\frac{1}{n^{2}}\,\int_{{\mathbb{T}}^{d}}|\sum_{j=0}^{n-1}e^{2\pi
i\langle
z_{j},{\underline{t}}\rangle}|^{2}\,d\nu_{\varphi}({\underline{t}})\leq\frac{M_{n}}{n^{2}}\,\int_{{\mathbb{T}}^{d}}|\sum_{j=0}^{n-1}e^{2\pi
i\langle
z_{j},{\underline{t}}\rangle}|^{2}\,d{\underline{t}}+\int_{\rho>M_{n}}\,\rho\,d{\underline{t}}$
$\displaystyle\leq$ $\displaystyle M_{n}{V_{n}\over
n^{2}}+(\lambda\\{\rho>M_{n}\\})^{\frac{1}{2}}\|\rho\|_{2}\leq
M_{n}{V_{n}\over n^{2}}+M_{n}^{-1}\,\|\rho\|_{2}^{2}.$

Taking $M_{n}=(\log n)^{1+\frac{1}{2}\eta}$, we obtain the bound

$\frac{1}{n^{2}}\,\|\sum_{j=0}^{n-1}U_{z_{j}}\|_{2}^{2}\,\leq{C\over(\log
n)^{1+\frac{1}{2}\eta}}$

and then we finish the proof as in 1).

Now, suppose that
$\displaystyle\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}|\langle
U_{\underline{\ell}},U_{{\underline{0}}}\rangle|^{\zeta}<\infty$ with
$1<\zeta<2$. The spectral density $\rho$ exists and is in $L^{2}(\lambda)$,
since $\displaystyle\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}|\langle
U_{\underline{\ell}},U_{{\underline{0}}}\rangle|^{2}<\infty$. Moreover it
belongs to $L^{\zeta^{\prime}}(\lambda)$ where $\zeta,\zeta^{\prime}$ are
conjugate exponents (see: [31], p. 102, or [21] Th. 31.22), and it satisfies:

$\|\rho\|_{\zeta^{\prime}}\leq(\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}|\langle
U_{\underline{\ell}},U_{0}\rangle|^{\zeta})^{1/\zeta}.$

Hölder’s inequality implies:
$\int_{\rho>M_{n}}\,\rho\,d{\underline{t}}\leq(\lambda\\{\rho>M_{n}\\})^{1/\zeta}\|\rho\|_{\zeta^{\prime}}$.
As

$\lambda\\{\rho>M_{n}\\}\leq
M_{n}^{-\zeta^{\prime}}\int\rho^{\zeta^{\prime}}\,d{\underline{t}}=M_{n}^{-\zeta^{\prime}}\|\rho\|_{\zeta^{\prime}}^{\zeta^{\prime}},$

it follows:

$\int_{\rho>M_{n}}\,\rho\,d{\underline{t}}\leq
M_{n}^{-\zeta^{\prime}/\zeta}\|\rho\|_{\zeta^{\prime}}^{1+\zeta^{\prime}/\zeta}.$

Therefore, we obtain

$\displaystyle\frac{1}{n^{2}}\,\int_{{\mathbb{T}}^{d}}|\sum_{j=0}^{n-1}e^{2\pi
i\langle
z_{j},{\underline{t}}\rangle}|^{2}\,d\nu_{\varphi}({\underline{t}})\leq
M_{n}{V_{n}\over n^{2}}+\int_{\rho>M_{n}}\,\rho\,d{\underline{t}}\leq
M_{n}{V_{n}\over
n^{2}}+M_{n}^{-\zeta^{\prime}/\zeta}\|\rho\|_{\zeta^{\prime}}^{1+\zeta^{\prime}/\zeta}.$

Now we take $M_{n}$ such that : $M_{n}/(\log
n)^{\beta}=M_{n}^{-\zeta^{\prime}/\zeta}$, i.e. $M_{n}=(\log
n)^{\beta/\zeta^{\prime}}$. We get

$\frac{1}{n^{2}}\,\|\sum_{j=0}^{n-1}U_{z_{j}}\|_{2}^{2}\,\leq{C\over(\log
n)^{\beta(1-1/\zeta^{\prime})}}={C\over(\log n)^{\beta/\zeta}}={C\over(\log
n)^{1+\eta/\zeta}}\text{ with }\eta>0,$

and the end of the proof is as above. ∎

######  Remarks 1.3.

1) Let us give an example of a non stationary r.f. $(U_{\underline{\ell}})$
which satisfies Condition 1) of the previous proposition.

We take
$(U_{\underline{\ell}}=V_{\underline{\ell}}\,W_{\underline{\ell}},{\underline{\ell}}\in{\mathbb{Z}}^{d})$,
where $(V_{\underline{\ell}})$ and $(W_{\underline{\ell}})$ are two r.f.’s
independent from each other, with $(V_{\underline{\ell}})$ centered stationary
and such that $\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}|\langle
V_{\underline{\ell}},V_{{\underline{0}}}\rangle|<\infty$, and
$(W_{\underline{\ell}})$ satisfying
$\sup_{{\underline{\ell}},{\underline{p}}}|\langle
W_{{\underline{\ell}}+{\underline{p}}},W_{{\underline{\ell}}}\rangle|<\infty$.

The r.f. $(W_{\underline{\ell}})$ can be viewed as a (multiplicative) noise
(which can be non stationary) independent from the r.f.
$(U_{\underline{\ell}})$. Clearly the condition in 1) is satisfied.

2) For a stationary r.f. $(U_{\underline{\ell}})$ with a bounded spectral
density (but with a series of correlations which may be not absolutely
summable), then like in 1) the condition $\beta>1$ is sufficient for the
conclusion of the theorem.

Now, we give a pointwise bound for the sampled sums, first for i.i.d. r.v.’s,
then for a stationary random field
$(U_{\underline{\ell}})_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}$ of associated
r.v.’s.

######  Proposition 1.4.

1) Suppose that the r.v.’s
$U_{\underline{\ell}},{\underline{\ell}}\in{\mathbb{Z}}^{d}$, are i.i.d.,
centered, uniformly bounded by a constant $K$,
$\|U_{\underline{0}}\|_{\infty}\leq K$, and that ${\mathbb{E}}|U_{0}|^{2}=1$.
Then it holds

(6) $\displaystyle\limsup_{n}\frac{|S_{n}|}{\sqrt{V_{n}}\,(2\log\log
n)^{\frac{1}{2}}}\leq K,\,{\mathbb{P}}{\text{-}a.e.}$

If $V_{n}=o(n^{2}\,(\log\log n)^{-1})$, then
$\displaystyle\lim_{n}\frac{S_{n}}{n}=0$, ${\mathbb{P}}$-a.e.

2) Suppose the random field stationary and the r.v.’s $U_{\underline{\ell}}$
centered associated.

a) For all $\varepsilon>0$, it holds, with
$\sigma_{n}:=\|\sum_{i=0}^{n-1}U_{z_{i}}\|_{2}$:

(7)
$\displaystyle\limsup_{n}\frac{|S_{n}|}{\sigma_{n}\,(\log\sigma_{n})^{\frac{1}{2}+\varepsilon}}\leq
1,\,{\mathbb{P}}{\text{-}a.e.}$

b) If moreover the r.f. has a summable series of correlations, then, for all
$\varepsilon>0$,

(8) $\displaystyle|S_{n}|=O(\sqrt{V_{n}}\,(\log
n)^{\frac{1}{2}+\varepsilon}),\,{\mathbb{P}}\text{-a.e.}$

If $\displaystyle V_{n}\leq Cn^{2}\,(\log n)^{-(1+\eta)})\text{ for some
constants }C,\eta>0$, then
$\displaystyle\lim_{n}\frac{S_{n}}{n}=0,\,{\mathbb{P}}\text{-a.e.}$

Proof. A) Recall that $\sigma_{n}=\|\sum_{i=0}^{n-1}U_{z_{i}}\|_{2}$. In case
2) we may assume $\|U_{{\underline{0}}}\|_{2}=1$, and then in all cases
$\sigma_{n}\leq n$ and by association $\sigma_{n}\geq n^{\frac{1}{2}}$. We
have in case 1) $\sigma_{n}=\sqrt{V_{n}}$ and in case 2b), for associated
variables, by (4): $\sigma_{n}\leq(\sum_{\underline{p}}\langle
U_{\underline{p}},U_{{\underline{0}}}\rangle)^{\frac{1}{2}}\,\sqrt{V_{n}}$. By
association, $\sigma_{n}$ is non-decreasing and tends to infinity.

For $\rho>1$, let $n_{k}=n_{k}(\rho)$ be a strictly increasing sequence of
integers such that $\rho^{k}<\sigma_{n_{k}}\leq\rho^{k+1}$. Since
$1\leq\sigma^{2}_{k+1}-\sigma^{2}_{k}\leq 1+2k$, such a sequence exists after
a certain rank. By the choice of $(n_{k})$ we have

(9)
$\displaystyle\rho^{k}<\sigma_{n_{k}}\leq\rho^{k+1}<\sigma_{n_{k+1}}\leq\rho^{k+2}.$

Moreover, we have $\sigma_{n_{k+1}}/\sigma_{n_{k}}\leq\rho^{2}$ and, since
$\sigma_{n}\leq n$, $n_{k}\geq\rho^{k}$. Let $(\lambda_{n})$ be a non
decreasing sequence of positive numbers such that

$\displaystyle\lambda_{n_{k}}>\sqrt{2},\
\limsup_{k}\lambda_{n_{k+1}}/\lambda_{n_{k}}\leq 1,$ (10)
$\displaystyle\sum_{k}{\mathbb{P}}\big{(}\big{|}\sum_{i=0}^{n_{k}-1}U_{z_{i}}\big{|}\geq\,(\lambda_{n_{k}}-\sqrt{2})\,\|\sum_{i=0}^{n_{k}-1}U_{z_{i}}\|_{2}\big{)}<\infty.$

By the previous inequalities and by Newman-Wright’s inequality (2) for the
sequence of centered associated random variables 111as it is a subset of a set
of associated r.v.’s $(W_{i})=(U_{z_{i}})$, we have

$\sum_{k}{\mathbb{P}}(\max_{0\leq j\leq
n_{k}-1}\big{|}\sum_{i=0}^{j}U_{z_{i}}\big{|}\geq\lambda_{n_{k}}\,\|\sum_{i=0}^{n_{k}-1}U_{z_{i}}\|_{2})\leq
2\sum_{k}{\mathbb{P}}(|\sum_{j=0}^{n_{k}-1}U_{z_{j}}|\geq(\lambda_{n_{k}}-\sqrt{2})\|\sum_{j=0}^{n_{k}-1}U_{z_{j}}\|_{2})<+\infty.$

By the Borel-Cantelli lemma, it follows:

$\limsup_{k}{\max_{0\leq j\leq
n_{k+1}-1}\big{|}\sum_{i=0}^{j}U_{z_{i}}\big{|}\over\,\lambda_{n_{k+1}}\,\sigma_{n_{k+1}}}\leq
1,{\mathbb{P}}\text{-a.e.}$

Hence ${\mathbb{P}}$-a.e.

(11) $\displaystyle\limsup_{k}\frac{\max_{0\leq
j<n_{k+1}-1}|\sum_{i=0}^{j}U_{z_{i}}|}{\lambda_{n_{k}}\sigma_{n_{k}}}\leq\limsup_{k}\bigl{(}\frac{\lambda_{n_{k+1}}}{\lambda_{n_{k}}}\frac{\sigma_{n_{k+1}}}{\sigma_{n_{k}}}\bigr{)}\leq\rho^{2}.$

Observe that, if $|S_{i}|>\rho^{2}\lambda_{i}\sigma_{i}$, for some
$i\in[n_{k},n_{k+1}[$, then $\max_{0\leq
j<n_{k+1}}|S_{j}|>\rho^{2}\lambda_{n_{k}}\sigma_{n_{k}}$. This shows:

$\displaystyle\\{|S_{n}|>\rho^{2}\lambda_{n}\sigma_{n},\,\text{i.o.}\\}\subset\\{\max_{0\leq
j<n_{k+1}}|S_{j}|>\rho^{2}\lambda_{n_{k}}\sigma_{n_{k}},\,\text{i.o.}\\}.$

By this inclusion and (11) it follows:
$\displaystyle\limsup_{n}\frac{|\sum_{i=0}^{n-1}U_{z_{i}}|}{\lambda_{n}\sigma_{n}}\leq\rho^{2},\,{\mathbb{P}}\text{-a.e.}$

Taking $\rho=\rho_{n}$ with $\rho_{n}\downarrow 1$, we obtain

(12)
$\displaystyle\displaystyle\limsup_{n}\frac{|\sum_{i=0}^{n-1}U_{z_{i}}|}{\lambda_{n}\sigma_{n}}\leq
1,\,{\mathbb{P}}\text{-a.e.}$

B) Choice of a sequence $(\lambda_{k})$ such that (10) is satisfied.

Case 1)

Suppose that the $U_{k}$’s are i.i.d. r.v.’s. Recall that if $(W_{j},j\geq 1)$
are centered bounded sequence of independent random variables on
$(\Omega,{\mathbb{P}})$, for any finite sum of the $W_{j}$’s it holds by
Hoeffding’s inequality for differences of martingale ([20]), for every
$\varepsilon>0$:

(13) $\displaystyle{\mathbb{P}}(|\sum_{j}W_{j}|>\varepsilon)\leq
2\exp(-\frac{1}{2}\frac{\varepsilon^{2}}{\sum_{j}\|W_{j}\|_{\infty}^{2}}).$

We apply it to the family
$(N_{n}({\underline{\ell}})U_{\underline{\ell}},\,{\underline{\ell}}\in{\mathbb{Z}}^{d})$.
From the hypotheses, we have:

$\sum_{\underline{\ell}}\|N_{n}({\underline{\ell}})U_{\underline{\ell}}\|_{\infty}^{2}\leq
K^{2}\sum_{\underline{\ell}}N_{n}^{2}({\underline{\ell}})=K^{2}V_{n}.$

With $\varepsilon=(\lambda-\sqrt{2})\sqrt{V_{n}}$, (13) implies:

$\displaystyle{\mathbb{P}}\big{(}\big{|}\sum_{\ell}N_{n}(\ell)\,U_{\underline{\ell}}\big{|}\geq(\lambda-\sqrt{2})\sqrt{V_{n}}\big{)}$
$\displaystyle\leq
2\exp\big{(}-\frac{1}{2}(\lambda-\sqrt{2})^{2}\frac{V_{n}}{K^{2}V_{n}}\big{)}=2\exp\big{(}-{1\over
2K^{2}}(\lambda-\sqrt{2})^{2}\big{)}.$

Let $c,\delta$ be such that $c>\delta>K^{2}$. In the previous inequality, we
take

$\lambda=\lambda_{n}=(2c\log\log n)^{\frac{1}{2}}.$

Let $k(c,\delta)$ be such that $\lambda_{n_{k}}>\sqrt{2}$ and
$c(1-\frac{2}{\sqrt{c\log\log n_{k}}})\geq\delta>1$, for $k\geq k(c,\delta)$.
We have:

$\displaystyle\sum_{k=k(c,\delta)}^{\infty}{\mathbb{P}}\big{(}\big{|}\sum_{i=1}^{n_{k}-1}U_{z_{i}}\big{|}\geq\,(\lambda_{n_{k}}-\sqrt{2})\,\|\sum_{i=1}^{n_{k}-1}U_{z_{i}}\|_{2}\big{)}\leq
2\sum_{k=k(c,\delta)}^{\infty}\exp\big{(}-{1\over
2K^{2}}{(\lambda_{n_{k}}-\sqrt{2})^{2}}\big{)}$
$\displaystyle\leq\frac{2}{\exp
K^{2}}\sum_{k=k(c,\delta)}^{\infty}\exp\big{(}-{c\over K^{2}}\log\log
n_{k})(1-\frac{2}{\sqrt{c\log\log n_{k}}})\big{)}$
$\displaystyle\leq\frac{2}{{\exp
K^{2}}}\sum_{k=k(c,\delta)}^{\infty}\frac{1}{(k\log\rho)^{\delta\over
K^{2}}}<\infty.$

Now we can apply (12). It follows:

$\limsup_{n}\frac{|\sum_{i=0}^{n-1}U_{z_{i}}|}{\sqrt{2c(\log\log n)V_{n}}}\leq
1,{\mathbb{P}}\text{-a.e.}$

Taking $c=c_{n}$ with $c_{n}\downarrow K^{2}$, we get (6).

Case 2)

For general associated r.v.’s, we use simply that
${\mathbb{P}}\big{(}\big{|}\sum_{i=0}^{n-1}U_{z_{i}}\big{|}\geq\,\lambda\,\|\sum_{i=0}^{n-1}U_{z_{i}}\|_{2}\big{)}\leq\frac{1}{\lambda^{2}}$.

We take $\lambda_{n}=(\log\sigma_{n})^{\frac{1}{2}+\varepsilon}$, with
$\varepsilon>0$. By (9) we have
$\lambda_{n_{k}}\geq(k\log\rho)^{\frac{1}{2}+\varepsilon}$, and therefore, for
a constant $C_{1}$: $\sum_{k}\frac{1}{\lambda_{n_{k}}^{2}}\leq
C_{1}\sum_{k}k^{-(1+2\varepsilon)}<+\infty$; hence condition (10).

Moreover we have
$k\log\rho\leq\log\sigma_{n_{k}}\leq\log\sigma_{n_{k+1}}\leq(k+2)\log\rho$;
hence

${\lambda_{n_{k+1}}\over\lambda_{n_{k}}}=\bigl{(}{\log\sigma_{n_{k+1}}\over\log\sigma_{n_{k})}}\bigr{)}^{\frac{1}{2}+\varepsilon}\leq(1+\frac{2}{k})^{\frac{1}{2}+\varepsilon}\to
1.$

By (12), this proves (7) in 2a)

For case 2b) we have $\sigma^{2}_{n}\leq V_{n}\sum_{\underline{p}}\langle
U_{\underline{p}},U_{{\underline{0}}}\rangle$ and $\sigma_{n}\leq n$, hence it
yields (8). The last conclusion in case 2b) is now clear. Remark that it
follows also from Proposition 1.2. ∎

###  1.2. A Glivenko-Cantelli type theorem

Empirical process

Let us consider a random field of r.v.’s
$(X_{\underline{\ell}},{\underline{\ell}}\in{\mathbb{Z}}^{d})$ on
$(\Omega,\mathcal{F},{\mathbb{P}})$ with common distribution function $F$. Let
$(z_{k})\subset\mathbb{Z}^{d}$ be a sequence with self-intersections
$(V_{n})$.

Notation. We say that
$(X_{\underline{\ell}},{\underline{\ell}}\in{\mathbb{Z}}^{d})$ satisfies a
Glivenko-Cantelli theorem along a sequence $(z_{k})$ in ${\mathbb{Z}}^{d}$ if

$\lim_{n}\sup_{s}|\frac{1}{n}\sum_{k=1}^{n}{\bf
1}_{(-\infty,s]}(X_{z_{k}}(\omega))-F(s)|=0,\text{ for
}{\mathbb{P}}\text{-a.e.}\omega.$

We show now a Glivenko-Cantelli theorem along a sequence $(z_{k})$ under
various hypotheses on $(z_{k})$ and on $(X_{\underline{\ell}})$ (mixing,
i.i.d., associated or PQD).

Le $(X_{\underline{\ell}},{\underline{\ell}}\in{\mathbb{Z}}^{d})$ be a r.f.
Denoting by $\sigma(X_{\underline{\ell}})$ the $\sigma$-algebra generated by
the random variable $X_{\underline{\ell}}$, we define a coefficient of mixing
by

(14)
$\displaystyle\gamma({\underline{\ell}}):=\sup_{r\in{\mathbb{Z}}^{d}}\sup_{A\in\sigma(X_{{\underline{r}}}),\,B\in\sigma(X_{{\underline{\ell}}+{\underline{r}}})}|{\mathbb{P}}(A\cap
B)-{\mathbb{P}}(A){\mathbb{P}}(B)|.$

Observe that for every $s,t\in{\mathbb{R}}$ it holds:

(15) $\displaystyle\sup_{{\underline{r}}\in{\mathbb{Z}}^{d}}|\langle
1_{X_{\underline{r}}\leq s}-{\mathbb{P}}(X_{\underline{r}}\leq
s),1_{X_{{\underline{\ell}}+{\underline{r}}}\leq
t}-{\mathbb{P}}(X_{{\underline{\ell}}+{\underline{r}}}\leq
t)\rangle|\leq\gamma({\underline{\ell}}),\forall{\underline{\ell}}\in{\mathbb{Z}}^{d}.$

By (15) and Proposition 1.2, we get:

######  Theorem 1.5.

Let $(z_{k})$ be such that $\displaystyle V_{n}\leq C_{1}{n^{2}\over(\log
n)^{\beta}}$, for constants $C_{1}>0,\beta$. If
$\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\gamma({\underline{\ell}})<+\infty$
and $\beta>1$, or if the r.f. is stationary and
$\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\gamma({\underline{\ell}})^{\zeta}<+\infty$,
for some $\zeta\in[1,2]$ and $\beta>\zeta$, then
$(X_{\underline{\ell}},{\underline{\ell}}\in{\mathbb{Z}}^{d})$ satisfies a
Glivenko-Cantelli theorem along $(z_{k})$.

Using Proposition 1.4, we consider now the i.i.d. and associated cases.

######  Theorem 1.6.

a) If $(X_{\underline{\ell}})_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}$ is a
r.f. of i.i.d. r.v.’s, then under the condition $V_{n}=o(n^{2}\,(\log\log
n)^{-1})$ it satisfies a Glivenko-Cantelli theorem along $(z_{k})$.

b) If $(X_{\underline{\ell}})_{{\underline{\ell}}\in\mathbb{Z}^{d}}$ is a
strictly stationary r.f. of associated r.v.’s such that
$\sum_{{\underline{\ell}}}\langle
X_{\underline{\ell}},X_{\underline{0}}\rangle$ converges, then, under the
condition $V_{n}=O(n^{2}\log^{-(1+\eta)}n)$ for some $\eta>0$, for a.e.
$\omega$, we have for each continuity point $s$ of $F$:

(16) $\displaystyle\lim_{n}\frac{1}{n}\sum_{k=0}^{n-1}{\bf
1}_{(-\infty,s]}(X_{z_{k}}(\omega))=F(s).$

If $F$ is continuous, the convergence is uniform in $s$.

Proof. a) Denote by $F_{n}(s)(\omega)$ the means
$\frac{1}{n}\sum_{k=0}^{n-1}{\bf 1}_{(-\infty,s]}(X_{z_{k}}(\omega))$. Let $Q$
be a dense countable set of continuity points of $F$.

For every $s\in Q$, by the assumption on $V_{n}$ and Proposition 1.4, there is
a null set $N(s)$ such that, for a sequence $\varepsilon_{n}$ tending to 0,
for every $\omega\not\in N(s)$,

$|F_{n}(s)(\omega)-F(s)|\leq\varepsilon_{n}(V_{n}\log\log
n)^{-\frac{1}{2}}|\sum_{k=0}^{n-1}{\bf\bigl{(}1}_{(-\infty,s]}(X_{z_{k}})-F(s)\bigr{)}|\to
0.$

Then $F_{n}(s)(\omega)\to F(s)$ for every $\omega$ outside the null set
$N:=\cup_{s\in Q}N(s)$ and for $s\in Q$.

Similarly by Proposition 1.4, for every $s$ in the set $J$ of jumps of $F$, we
have $F_{n}(s)(\omega)-F_{n}(s^{-})(\omega)\to F(s)-F(s^{-})$ a.e. As $J$ is
countable, this convergence holds for every $s\in J$ and
$\omega\not\in\tilde{N}$, where $\tilde{N}$ is a null set.

Outside the null set $N\cup\tilde{N}$, Lemma 1.1 applied with $Q$ and $J$
implies the result.

b) We consider now the case of a strictly stationary random field of
associated r.v.’s. Let $s$ be a continuity point of the common distribution
$F$. For every $\epsilon>0$ there exists $\delta>0$, such that
$F(s+\delta)-F(s-\delta)\leq\epsilon$. As in [30], for $\delta>0$ and $s$,
define the approximated step function $h_{\delta,s}$ by $h_{\delta,s}(x)=0$,
if $x\leq s-\delta$ and $h_{\delta,s}(x)=1+\frac{x-s}{\delta}$ if
$s-\delta\leq x\leq s$, otherwise, $h_{\delta,s}(x)=1$. It is a non decreasing
continuous function with $h^{\prime}_{\delta,s}(x)=1/\delta$ for
$s-\delta<x<s$. It follows from the above Hoeffding’s identity applied to this
approximated step function (see [2]):

$\displaystyle{\text{C}ov}(h_{\delta,s}(X_{\underline{\ell}}),h_{\delta,s}(X_{\underline{0}}))\leq\delta^{-2}\langle
X_{\underline{\ell}},X_{\underline{0}}\rangle,$
$\displaystyle{\text{C}ov}(h_{\delta,s+\delta}(X_{\underline{\ell}}),h_{\delta,s+\delta}(X_{\underline{0}}))\leq\delta^{-2}\langle
X_{\underline{\ell}},X_{\underline{0}}\rangle.$

By association and non decreasing,
$\big{(}h_{\delta,s}(X_{\underline{\ell}})\big{)}$ as well as
$\big{(}h_{\delta,s+\delta}(X_{\underline{\ell}})\big{)}$ are stationary r.f.s
of associated r.v.’s, and we may apply Proposition 1.4 to their centered
versions (also associated). The condition simply reads, for $\tau=s,s+\delta$:

$\sum_{{\underline{\ell}}}{\text{C}ov}(h_{\delta,\tau}(X_{\underline{\ell}}),h_{\delta,\tau}(X_{\underline{0}}))\leq\delta^{-2}\sum_{\underline{\ell}}\langle
X_{\underline{\ell}},X_{\underline{0}}\rangle<\infty.$

We put $\overline{S}_{n}=\sum_{k=0}^{n-1}h_{\delta,s}(X_{z_{k}})$ and
$\underline{S}_{n}=\sum_{k=0}^{n-1}h_{\delta,s+\delta}(X_{z_{k}})$. By
$h_{\delta,s+\delta}(x)\leq{\bf 1}_{\\{x>s\\}}\leq h_{\delta,s}(x)$, it holds
$\underline{S}_{n}\leq\sum_{k=0}^{n-1}{\bf
1}_{(s,\infty)}(X_{z_{k}})\leq\overline{S}_{n}$. Hence by Proposition 1.4, we
have almost everywhere
$\frac{1}{n}\underline{S}_{n}\to{\mathbb{E}}[h_{\delta,s+\delta}(X_{\underline{0}})]\
\text{and}\
\frac{1}{n}\overline{S}_{n}\to{\mathbb{E}}[h_{\delta,s}(X_{\underline{0}})]$.
Since

$\displaystyle{\mathbb{E}}[h_{\delta,s}(X_{\underline{0}})]\leq
F(s)-F(s-\delta)+1-F(s)\leq\epsilon+1-F(s),$
$\displaystyle{\mathbb{E}}[h_{\delta,s+\delta}(X_{\underline{0}})]\geq
1-F(s+\delta)=1-F(s)-(F(s+\delta)-F(s))\geq 1-F(s)-\epsilon,$

we conclude

$1-F(s)-\epsilon\leq\liminf_{n}\frac{1}{n}\sum_{k=0}^{n-1}{\bf
1}_{(s,\infty)}(X_{z_{k}})\leq\limsup_{n}\frac{1}{n}\sum_{k=0}^{n-1}{\bf
1}_{(s,\infty)}(X_{z_{k}})\leq 1-F(s)+\epsilon.$

Subtracting the $1$’s and taking $\epsilon\to 0$, we get (16). ∎

PQD variables.

The result shown for associated variables can be extended to the class of PDQ
variables, but with a stronger condition on the local times of the sequence
$(z_{k})$.

######  Proposition 1.7.

Let $(U_{\underline{\ell}})$ be a centered stationary random field of pairwise
PQD r.v.’s such that $\sum_{\underline{\ell}}\langle
U_{{\underline{\ell}}},U_{\underline{0}}\rangle$ converges. Let $(z_{k})$ be a
sequence of points with maximal local times sequence $(M_{n})$. If
$\sum_{n\geq 1}{M_{n}\over n^{2}}<+\infty$, then
$\frac{1}{n}(U_{z_{0}}+\cdots+U_{z_{n-1}})$ converges a.e. to $0$.

Proof. We apply the following result of [4]: let $(Y_{j}:j\geq 1)$ be a
sequence of pairwise centered PQD r.v.’s. with finite variance. If
$\sum_{j\geq 1}j^{-2}\text{Cov}(Y_{j},\sum_{i=1}^{j}Y_{i})$ converges and
$\sup_{j}{\mathbb{E}}|Y_{j}|<\infty$, then $n^{-1}\sum_{i=1}^{n}Y_{i}\to 0$
a.e.

Taking for $Y_{j}$ the (still) pairwise PQD r.v.’s $U_{z_{j}}$, we get the
result, since $\text{Cov}(U_{z_{j}},U_{z_{1}}+\cdots+U_{z_{j}})\leq
M_{j}\sum_{\underline{\ell}}\langle
U_{{\underline{0}}},U_{\underline{\ell}}\rangle$. ∎

Now, we consider the empirical distribution.

######  Theorem 1.8.

Let $(X_{\underline{\ell}})$ be a centered strictly stationary random field of
pairwise PQD r.v.’s with distribution function $F$ such that
$\sum_{\underline{\ell}}\langle
X_{{\underline{\ell}}},X_{\underline{0}}\rangle$ converges. Let $(z_{k})$ be a
sequence of points with maximal local times sequence $(M_{n})$. If
$\sum_{n\geq 1}{M_{n}\over n^{2}}<+\infty$, then for each continuity point $s$
of $F$, we have for a.e. $\omega$: $\lim_{n}\frac{1}{n}\sum_{k=0}^{n-1}{\bf
1}_{(-\infty,s]}(X_{z_{k}}(\omega))=F(s)$.

In particular, if $F$ is continuous, the above convergence is uniform over
$s$.

Proof. The r.f.s $h_{\delta,s}(X_{\underline{\ell}})$ and
$h_{\delta,s+\delta}(X_{\underline{\ell}})$ are still stationary pairwise PQD.
The proof is analogous to the proof of Theorem 1.6. For the last statement, we
use Lemma 1.1. ∎

Remark. If $M_{n}=O(n\,(\log n)^{-(1+\eta)})$, then $V_{n}=O(n^{2}\,(\log
n)^{-(1+\eta)})$.

If $\displaystyle V_{n}\leq C{n^{2}\over(\log n)^{\beta}}$, with $\beta>2$,
then $\displaystyle\sum_{n\geq 1}{M_{n}\over n^{2}}<+\infty$.

As shown in Section 3, $\sum_{n\geq 1}{M_{n}\over n^{2}}$ converges when the
sampling is done along random walks, but diverges in some examples of sampling
along “deterministic” random walks.

###  1.3. A sufficient condition for a FCLT for the sampled empirical process

After a Glivenko-Cantelli theorem for sampled empirical processes, we consider
now the Functional Central Limit Theorem (FCLT). Let $(z_{k})$ be in
$\mathbb{Z}^{d},d\geq 1,$ with the associated quantities
$N_{n}({\underline{\ell}})$, $M_{n}$ and $V_{n}$ defined by (1).

Before restricting to a r.f. of i.i.d. r.v.’s, first we examine the variance
in the more general situation where the series of correlations is absolutely
summable.

Kernel associated to a sequence $(z_{k})$ and variance.

Let $K_{n}$ be the kernel (which is a real even function on ${\mathbb{T}}^{d}$
depending on $n\geq 0$) defined by the equivalent formulas:

(17) $\displaystyle K_{n}({\underline{t}})$ $\displaystyle=$
$\displaystyle|\sum_{k=0}^{n-1}e^{2\pi i\langle
z_{k},{\underline{t}}\rangle}|^{2}=n+2\sum_{k=1}^{n-1}\,\sum_{j=0}^{n-k-1}\cos(2\pi\langle
z_{k+j}-z_{j},t\rangle)=|\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}N_{n}({\underline{\ell}})\,e^{2\pi
i\langle{\underline{\ell}},t\rangle}|^{2}$ $\displaystyle=$ $\displaystyle
n+2\sum_{\underline{\ell}}\bigl{(}\sum_{k=1}^{n-1}\,\sum_{j=0}^{n-k-1}1_{z_{k+j}-z_{j}={\underline{\ell}}}\bigr{)}\cos(2\pi\langle{\underline{\ell}},t\rangle).$

If $(X_{\underline{\ell}},{\underline{\ell}}\in{\mathbb{Z}}^{d})$ is a
stationary r.f. such that
$\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}|\langle
X_{\underline{\ell}},X_{\underline{0}}\rangle|<+\infty$, the spectral density
$\rho$ is continuous and we have:

$\int|\sum_{k=0}^{n-1}X_{z_{k}}|^{2}d{\mathbb{P}}=\int_{{\mathbb{T}}^{d}}K_{n}({\underline{t}})\,\rho({\underline{t}})\,d{\underline{t}}\leq\|\rho\|_{\infty}V_{n}\leq(\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}|\langle
X_{\underline{\ell}},X_{\underline{0}}\rangle|)V_{n}.$

One can ask if there is an asymptotic variance, i.e., a limit for the
normalised quantity $\displaystyle
V_{n}^{-1}\int|\sum_{k=0}^{n-1}X_{z_{k}}|^{2}d{\mathbb{P}}$ which is bounded
if the series of correlations is absolutely summable.

The existence of asymptotic variance is shown in [11] in the case of summation
along a random walk. We will come back to the question of its positivity for
transient random walks in Subsection 3.1.

Functional Central limit Theorem in the i.i.d. case

The following result gives a sufficient condition for a Functional Central
limit Theorem (FCLT) along a sequence $(z_{k})$ in the i.i.d. case.

The standard Brownian bridge process $W^{0}(s)$ is the centered Gaussian
process $W^{0}(s):=W(s)-sW(1)$ in $C(0,1)$, where $W(s)$ is the Wiener
process. It has the properties $W^{0}(0)=W^{0}(1)=0$ and
${\mathbb{E}}[W^{0}(s_{1})W^{0}(s_{2})]=s_{1}\wedge s_{2}-s_{1}s_{2}$.

Let $(X_{\underline{k}})_{{\underline{k}}\in\mathbb{Z}^{d}}$ be i.i.d. random
variables with common probability distribution $F$ in $[0,1]$. We put
$W_{F}^{0}=W^{0}\circ F$. Let $Y_{n}(s)$ be the random element in $D[0,1]$
defined by

$\displaystyle Y_{n}(s)=\frac{1}{\sqrt{V_{n}}}\sum_{k=0}^{n-1}\,[{\bf
1}_{X_{z_{k}}\leq
s}-F(s)]=\frac{1}{\sqrt{V_{n}}}\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}N_{n}(\ell)\,[{\bf
1}_{X_{{\underline{\ell}}}\leq s}-F(s)].$

######  Theorem 1.9.

$Y_{n}(s)\to_{D[0,1]}W_{F}^{0}(s)$, if $(z_{k})$ satisfies the condition

(18) $\displaystyle\lim_{n}{M^{2}_{n}\over V_{n}}=0,$

Proof. The result follows from the Cramér-Wold device, if we prove convergence
of the finite dimensional distributions and tightness. The variance is

(19)
$\displaystyle{\mathbb{E}}[Y_{n}(s)]^{2}=\frac{1}{{V_{n}}}\,\sum_{{\underline{\ell}}}\,N_{n}^{2}(\ell)\,{\mathbb{E}}[{\bf
1}_{X_{\underline{\ell}}\leq s}-F(s)]^{2}=\sigma^{2}(s)=F(s)(1-F(s)).$

1) Finite dimensional distributions. The convergence follows from Lindeberg’s
theorem for triangular arrays of independent random variables as in [3, thm
7.2]. The Lindeberg’s condition for the triangular array of independent r.v.’s
$\displaystyle\big{(}\frac{N_{n}(\ell)[{\bf 1}_{X_{\underline{\ell}}\leq
s}-F(s)]}{\sqrt{V_{n}}}\big{)}_{{\underline{\ell}},n}$ follows from

$\displaystyle\frac{1}{V_{n}}\sum_{\underline{\ell}}\int_{\\{N_{n}({\underline{\ell}})|{\bf
1}_{X_{\underline{\ell}}\leq
s}-F(s)|\geq\,\varepsilon\sqrt{V_{n}}\\}}N_{n}^{2}({\underline{\ell}})\,|{\bf
1}_{X_{\underline{\ell}}\leq s}-F(s)|^{2}d{\mathbb{P}}$ $\displaystyle\leq$
$\displaystyle\frac{1}{V_{n}}\sum_{\underline{\ell}}N_{n}^{2}({\underline{\ell}})\int_{\\{\sup_{\underline{\ell}}N_{n}({\underline{\ell}})|{\bf
1}_{X_{\underline{0}}\leq s}-F(s)|\geq\varepsilon\sqrt{V_{n}}\\}}\,|{\bf
1}_{X_{\underline{0}}\leq s}-F(s)|^{2}d{\mathbb{P}}\to 0,$

for every $\varepsilon>0$, since
$V_{n}=\sum_{\underline{\ell}}N_{n}^{2}({\underline{\ell}})$ and
$\displaystyle{\sup_{\underline{\ell}}N_{n}({\underline{\ell}})\over\sqrt{V_{n}}}\to
0$, by assumption.

For the correlation of the process taken at $s_{1}$ and $s_{2}$, it holds by
independence:

$\displaystyle{\mathbb{E}}[Y_{n}(s_{1})Y_{n}(s_{2})]$ $\displaystyle=$
$\displaystyle\frac{1}{V_{n}}\sum_{{\underline{\ell}}_{1},{\underline{\ell}}_{2}}N_{n}({\underline{\ell}}_{1})N_{n}({\underline{\ell}}_{2}){\mathbb{E}}[({\bf
1}_{X_{{\underline{\ell}}_{1}}\leq s_{1}}-F(s_{1}))({\bf
1}_{X_{{\underline{\ell}}_{2}}\leq s_{2}}-F(s_{2}))]$ $\displaystyle=$
$\displaystyle\frac{1}{V_{n}}\sum_{\ell}N^{2}_{n}({\underline{\ell}})(F(s_{1}\wedge
s_{2})-F(s_{1})F(s_{2}))=F(s_{1}\wedge s_{2})-F(s_{1})F(s_{2}).$

This proves the convergence in distribution: $Y_{n}(s)\to W_{F}^{0}(s)$ for
every $s$.

Let us show now the convergence of the finite dimensional distributions.
Starting with the asymptotic distribution of $aY_{n}(s_{1})+bY_{n}(s_{2})$, by
the above computation, we have

$\displaystyle{\mathbb{E}}[(aY_{n}(s_{1})+bY_{n}(s_{2}))^{2}]=$ (20)
$\displaystyle
a^{2}F(s_{1})(1-F(s_{1}))+b^{2}F(s_{2})(1-F(s_{2}))+2ab(F(s_{1}\wedge
s_{2})-F(s_{1})F(s_{2})).$

As above, it is easily seen that Lindeberg’s condition is satisfied for the
triangular array defined by $aY_{n}(s_{1})+bY_{n}(s_{2})$. It means that the
asymptotic distribution of $aY_{n}(s_{1})+bY_{n}(s_{2})$ is centered Gaussian
with variance as computed above.

Note that ${\mathbb{E}}[(aW^{0}(s_{1})+bW^{0}(s_{2}))^{2}]$ is also given by
(20) above.

Similarly, for every $s_{1}\leq\cdots\leq s_{r}$, it holds

$(Y_{n}(s_{1}),\cdots,Y_{n}(s_{r}))\to_{dist}(W_{F}^{0}(s_{1}),\ldots,W_{F}^{0}(s_{r})).$

Tightness. First we suppose $F$ continuous. Following the method of [3], it is
enough to show that for $s\leq t\leq r$, uniformly in $n$,

$\displaystyle{\mathbb{E}}[(Y_{n}(t)-Y_{n}(s))^{2}(Y_{n}(r)-Y_{n}(t))^{2}]\leq
C(F(r)-F(s))^{2}.$

Putting $F(u,v):=F(v)-F(u)$, $f({\underline{\ell}},u,v):={\bf
1}_{u<X_{\underline{\ell}}\leq v}-F(u,v)$, for $u<v$, we compute

$\displaystyle{\mathbb{E}}[(Y_{n}(t)-Y_{n}(s))^{2}(Y_{n}(r)-Y_{n}(t))^{2}]=\frac{1}{V^{2}_{n}}{\mathbb{E}}\big{[}\big{(}\sum_{{\underline{\ell}}}N_{n}({\underline{\ell}})f({\underline{\ell}},s,t)\big{)}^{2}\big{(}\sum_{{\underline{\ell}}}\,N_{n}({\underline{\ell}})f({\underline{\ell}},t,r)\big{)}^{2}\big{]}.$

By expansion and independence, the above expression is sum of three types of
terms:

$\frac{1}{V^{2}_{n}}\sum_{\underline{\ell}}N_{n}^{4}({\underline{\ell}})\,[A],\,\frac{1}{V^{2}_{n}}\sum_{{\underline{\ell}}_{1},{\underline{\ell}}_{2}}N_{n}^{2}({\underline{\ell}}_{1})N_{n}^{2}({\underline{\ell}}_{2})\,[B],\,\frac{1}{V^{2}_{n}}\sum_{{\underline{\ell}}_{1}\not={\underline{\ell}}_{2}}N_{n}^{2}({\underline{\ell}}_{1})N_{n}^{2}({\underline{\ell}}_{2})\,[C],$
with $\displaystyle
A={\mathbb{E}}[f^{2}({\underline{\ell}},s,t)f^{2}({\underline{\ell}},t,r)],B={\mathbb{E}}[f^{2}({\underline{\ell}}_{1},s,t)]\,{\mathbb{E}}[f^{2}({\underline{\ell}}_{2},t,r)],$
$\displaystyle
C={\mathbb{E}}[f({\underline{\ell}}_{1},s,t)f({\underline{\ell}}_{1},t,r))]\,{\mathbb{E}}[f({\underline{\ell}}_{2},s,t)f({\underline{\ell}}_{2},t,r))].$

By stationarity and since the intervals do not overlap, we have

$\displaystyle A$
$\displaystyle=F(s,t)F^{2}(t,r)+F^{2}(s,t)F(t,r)-3F^{2}(s,t)F^{2}(t,r),$
$\displaystyle B$ $\displaystyle=F(s,t)(1-F(s,t))\cdot
F(t,r)(1-F(t,r)),\,C=F^{2}(s,t)F^{2}(t,r).$

Since $0\leq F(s,t),F(t,r),F(s,r)\leq 1$ and $F(s,t),F(t,r)\leq F(s,r)$, it
follows

$\displaystyle A\leq 2F^{3}(s,r)\leq 2F^{2}(s,r),\,B\leq F^{2}(s,r),\ C\leq
F^{4}(s,r)\leq F^{2}(s,r).$

Recall that $V_{n}=\sum_{\underline{\ell}}N_{n}^{2}({\underline{\ell}})$.
Using $\|\cdot\|_{\ell_{4}}\leq\|\cdot\|_{\ell_{2}}$ for the bound of the
first term, we have for some fixed constant $C>0$:

$\displaystyle{\mathbb{E}}[(Y_{n}(t)-Y_{n}(s))^{2}(Y_{n}(r)-Y_{n}(t))^{2}]\leq
C(F(r)-F(s))^{2},\ \forall n.$

Hence by [3, Theorem 15.6], for non decreasing continuous $F$, the sequence of
processes $(Y_{n}(s))$ is tight in $D(0,1)$. This proves that, if $F$ is
continuous, then $Y_{n}\to_{D(0,1)}(W^{0}\circ F)$.

Now, for a general $F$ a classical method is to use a generalized inverse. Let
us describe it briefly. We consider first the uniform empirical process. Let
$(\zeta_{k})$ be uniformly distributed i.i.d. r.v.’s. Denote the empirical
process along $(z_{k})$ with respect to $(\zeta_{k})$ by $U_{n}(s)$. By
applying what we have just proved for a continuous distribution,
$U_{n}(s)\to_{D(0,1)}W^{0}(s)$.

Now let $F^{-1}(t):=\inf\\{s:t\leq F(s)\\}$. We get
${\mathbb{P}}(F^{-1}(\zeta_{0})\leq s)={\mathbb{P}}(X_{0}\leq s)=F(s)$, so
$Y_{n}(s)=_{dist.}U_{n}(F(s))$. We may then proceed as in Billingsley ([3,
Theorem 5.1]) to deduce the FCLT for $Y_{n}(s)$ with $W^{0}(F(s))$ as limit. ∎

##  2\. Local times for ergodic sums

In the previous section about limit theorems for the empirical process sampled
along $(z_{k})$, we have found sufficient conditions on the quantities $V_{n}$
and $M_{n}$ associated to $(z_{k})$. When $(z_{k})$ is given by a “cocycle”,
$z_{k}=S_{k}f(x)$, one can ask if these conditions are satisfied.

We start with some general facts and construct counterexamples for which
condition (18) is not satisfied. In the next section, we will discuss two very
different examples of cocycles: first the case of random walks, then
“stationary random walks” generated by a rotation.

###  2.1. Auxiliary general results

First we introduce some notation and make general remarks.

######  Notation 2.1.

Let $(X,{\cal B},\mu)$ be a probability space and $T$ a measure preserving
transformation acting on $X$ such that the dynamical system $(X,{\cal
B},\mu,T)$ is ergodic.

Let $f$ be a measurable function on $X$ with values in ${\mathbb{Z}}^{d}$,
$d\geq 1$. Its ergodic sums generated by the iteration of $T$, denoted by
$f_{k}$ (or $S_{k}f$), are

$f_{k}(x):=\sum_{j=0}^{k-1}f(T^{j}x),k\geq 1,\ f_{0}(x)=0.$

The sequence $(f_{k}(x),k\geq 1)$ can be viewed as a “stationary random walk”
defined on $(X,{\cal B},\mu)$. It will be called a “cocycle” and denoted by
$(\mu,T,f)$ or simply $(T,f)$.

For $x\in X$, we put (cf. (1)) $N_{0}(x,{\underline{\ell}})=0$ and, for $n\geq
1$,

$\displaystyle N_{n}(T,f,x,{\underline{\ell}})$ $\displaystyle:=$
$\displaystyle\\#\\{1\leq k\leq n:\
f_{k}(x)={\underline{\ell}}\\},\,{\underline{\ell}}\in{\mathbb{Z}}^{d},$
$\displaystyle M_{n}(T,f,x)$ $\displaystyle:=$
$\displaystyle\max_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}N_{n}(T,f,x,{\underline{\ell}}),$
$\displaystyle V_{n}(T,f,x)$ $\displaystyle:=$ $\displaystyle\\#\\{1\leq
j,k\leq
n:\,f_{j}(x)=f_{k}(x)\\}=\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\,N_{n}^{2}(x,{\underline{\ell}}).$

Most of the time, we will omit $T$ and $f$ in the notation and write simply
$N_{n}(x,{\underline{\ell}})$, $M_{n}(x)$, $V_{n}(x)$. We have
$\sum_{\underline{\ell}}N_{n}(x,{\underline{\ell}})=n\text{ and }n\leq
V_{n}(x)\leq n\,M_{n}(x)$.

A question is to know if the following conditions hold for a.e. $x$:

(21) $\displaystyle V_{n}(x)=o(n^{2}\,(\log\log n)^{-1})\text{ or
}V_{n}(x)\leq C_{1}{n^{2}\over(\log n)^{\beta}},\text{ with }\beta>1,$ (22)
$\displaystyle\lim_{n}{M^{2}_{n}(x)\over V_{n}(x)}=0.$

For a random walk this is related to a question studied in [16] and later in
[15]: How many times does the walk revisit the most frequently visited site in
the first $n$ steps?

Cylinder map. We denote by $\tilde{T}_{f}$ the map (sometimes called cylinder
map) $(x,{\underline{\ell}})\to(Tx,{\underline{\ell}}+f(x))$ acting on
$X\times{\mathbb{Z}}^{d}$, endowed with the infinite invariant measure
$\tilde{\mu}$ defined as the product of $\mu$ by the counting measure on
${\mathbb{Z}}^{d}$.

For $\varphi:X\times{\mathbb{Z}}^{d}\to{\mathbb{R}}$ the ergodic sums for the
cylinder map are

$\tilde{S}_{n}\varphi(x,{\underline{\ell}}):=\sum_{k=0}^{n-1}\varphi(\tilde{T}_{f}^{k}(x,{\underline{\ell}}))=\sum_{k=0}^{n-1}\varphi(T^{k}x,{\underline{\ell}}+f_{k}(x)).$

With $\varphi_{0}:={\bf 1}_{X\times\\{{\underline{0}}\\}}$, it holds

$\tilde{S}_{n}\varphi_{0}(x,-{\underline{\ell}})=\sum_{k=0}^{n-1}{\bf
1}_{X\times\\{{\underline{0}}\\}}(T^{k}x,-{\underline{\ell}}+f_{k}(x))=\\#\\{0\leq
k\leq n-1:\,f_{k}(x)={\underline{\ell}}\\}.$

Therefore,
$\tilde{S}_{n}\varphi_{0}(x,-{\underline{\ell}})=N_{n}({\underline{\ell}})(x)$.

Recurrence/transience. It can be shown that a cocycle $(\mu,T,f)$ (over an
ergodic dynamical system) is either recurrent or transient. For $f$ with
values in ${\mathbb{Z}}^{d}$, it means that either $S_{k}f(x)=0$ infinitely
often for a.e. $x$, or $S_{k}f(x)=0$ finitely often for a.e. $x$. In the
latter case, we have $\lim_{k}|S_{k}f(x)|=+\infty$, $\mu$-a.e.

Let ${\mathcal{R}}_{n}(x)=\\{{\underline{\ell}}\in{\mathbb{Z}}^{d}:\
f_{k}(x)={\underline{\ell}}\text{ for some }k\leq n\\}$ be the “range” of the
cocycle, i.e., the set of points visited by $f_{k}(x)$ up to time $n$.

In [14] the following result is shown (for the general case of a cocycle with
values in a countable group): let $G$ be a countable group and $f:X\to G$. If
the cocycle $(T,f)$ is recurrent, then ${\rm Card}({\mathcal{R}}_{n}(x))=o(n)$
for a.e. $x$. If it is transient, there exists $c>0$ such that ${\rm
Card}({\mathcal{R}}_{n}(x))\sim c\,n$ for a.e. $x$.

Using the lemma below, this implies for a.e. $x$:

(23) $\displaystyle\liminf_{n}{V_{n}(x)\over n}>0\text{ in the transient
case},\ {V_{n}(x)\over n}\to+\infty\text{ in the recurrent case}.$

To show (23) we use the following inequality valid for a general sequence
$(z_{k})$:

######  Lemma 2.2.

If $\mathcal{A}$ is a non empty subset in ${\mathbb{Z}}^{d}$, we have:

(24) $\displaystyle
V_{n}\geq{\bigl{(}\sum_{k=0}^{n-1}1_{z_{k}\,\in\mathcal{A}}\bigr{)}^{2}\over{\rm
Card}(\mathcal{A})}.$

Proof. Cauchy-Schwarz inequality implies:

$\displaystyle\sum_{k=0}^{n-1}1_{z_{k}\,\in\mathcal{A}}=\sum_{{\underline{\ell}}\in\mathcal{A}}\sum_{k=0}^{n-1}1_{z_{k}\,={\underline{\ell}}}\leq(\sum_{{\underline{\ell}}\in
A}(\sum_{k=0}^{n-1}1_{z_{k}\,={\underline{\ell}}})^{2})^{\frac{1}{2}}\,({\rm
Card}(\mathcal{A}))^{\frac{1}{2}}\leq V_{n}^{\frac{1}{2}}\,\,({\rm
Card}(\mathcal{A}))^{\frac{1}{2}}.\ \qed$

If $z_{k}=S_{k}f(x)$, this show (23). Indeed by taking
$\mathcal{A}=\mathcal{R}_{n}(x)$ we get

(25) $\displaystyle V_{n}(x)\geq{n^{2}\over{\rm Card}(\mathcal{R}_{n}(x))}.$

######  Lemma 2.3.

The following formulas hold for quantities defined in 2.1.

(26) $\displaystyle V_{n}(x)$ $\displaystyle=$ $\displaystyle
n+2\sum_{k=1}^{n-1}\,\sum_{j=0}^{n-k-1}(1_{f_{k}(T^{j}x)={\underline{0}}}),$
(27) $\displaystyle=$ $\displaystyle
2[N_{n-1}(Tx,0)+N_{n-2}(T^{2}x,0)+...+N_{1}(T^{n-1}x,0)]+n,\ n\geq 2,$ (28)
$\displaystyle M_{n}(x)$ $\displaystyle=$
$\displaystyle\max[N_{n}(x,0),1+\max_{1\leq k\leq n-1}N_{n-k}(T^{k}x,0)]\leq
1+\max_{0\leq k\leq n-1}N_{n}(T^{k}x,0),$ (29) $\displaystyle=$ $\displaystyle
M_{n-1}(Tx)+1_{{\underline{\ell}}(n-1,Tx)={\underline{0}}}\leq M_{n-1}(Tx)+1.$

Proof. a) From $f_{k}(x)=f(x)+f_{k-1}(Tx),\,k\geq 1$, it follows

(30) $\displaystyle
N_{n}(x,{\underline{\ell}})=N_{n-1}(Tx,{\underline{\ell}}-f(x))+1_{f(x)={\underline{\ell}}},\,n\geq
1.$

Therefore we have:

$\displaystyle\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}N_{n}^{2}(x,{\underline{\ell}})=\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}[N_{n-1}(Tx,{\underline{\ell}}-f(x))+1_{f(x)={\underline{\ell}}}]^{2}=\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}[N_{n-1}(Tx,{\underline{\ell}})+1_{{\underline{\ell}}={\underline{0}}}]^{2}$
$\displaystyle=\sum_{{\underline{\ell}}\not=0}[N_{n-1}(Tx,{\underline{\ell}})]^{2}+[N_{n-1}(Tx,{\underline{0}})+1]^{2}=\sum_{{\underline{\ell}}}[N_{n-1}(Tx,{\underline{\ell}})]^{2}+2N_{n-1}(Tx,{\underline{0}})+1.$

Hence the relation

(31) $\displaystyle V_{n}(x)=V_{n-1}(Tx)+2N_{n-1}(Tx,{\underline{0}})+1.$

We have $V_{1}(x)=1$ and by the previous relation we get by induction (26) and
(27).

b) For $x\in X$, let ${\underline{\ell}}(n,x)$ (a most visited site by
$S_{k}(x)$ up to time $n$) be defined by

$\displaystyle{\underline{\ell}}(n,x)$ $\displaystyle:=$
$\displaystyle{\underline{0}},\text{ if }N_{n}(x,{\underline{0}})\geq
N_{n}(x,{\underline{\ell}}),\text{ for all }{\underline{\ell}}\not=0,$ else
$\displaystyle:=$ $\displaystyle{\underline{\ell}}_{1},\text{ if
}\ell_{1}\text{ is such that
}M_{n}(x)=N_{n}(x,{\underline{\ell}}_{1})>N_{n}(x,{\underline{0}}).$

Let $p_{n}(x)\in[1,n]$ be the first visit of $S_{k}(x)$ to
${\underline{\ell}}(n,x)$ for $k=1,...,n$. By definition
$M_{n}(x)=N_{n}(x,{\underline{\ell}}(n,x))$.

We have $M_{n}(x)=N_{n}(x,0)$ if ${\underline{\ell}}(n,x)=0$, else
$M_{n}(x)=N_{n-p_{n}(x)}(T^{p_{n}(x)}x,0)+1$, by the cocycle relation
$S_{p_{n}(x)+k}(x)-S_{p_{n}(x)}(x)=S_{k}(T^{p_{n}(x)}x)$. This implies:

$M_{n}(x)\leq\max[N_{n}(x,0),N_{n-p_{n}(x)}(T^{p_{n}(x)}x,0)+1]\leq\max[N_{n}(x,0),\max_{1\leq
k\leq n}N_{n-k}(T^{k}x,0)+1].$

It follows (noticing that $N_{0}(x,0)=0$):

(32) $\displaystyle M_{n}(x)\leq 1+\max_{0\leq k\leq n-1}N_{n-k}(T^{k}x,0)\leq
1+\max_{0\leq k\leq n-1}N_{n}(T^{k}x,0).$

This shows (28).

c) Observe also the following relation: by (30) we have:

$\displaystyle M_{n}(x)$ $\displaystyle=$
$\displaystyle\sup_{\underline{\ell}}[N_{n-1}(Tx,{\underline{\ell}}-f(x))+1_{{\underline{\ell}}-f(x)={\underline{0}}}]=\sup_{\underline{\ell}}[N_{n-1}(Tx,{\underline{\ell}})+1_{{\underline{\ell}}={\underline{0}}}]$
$\displaystyle=$
$\displaystyle\max\,[\sup_{{\underline{\ell}}\not={\underline{0}}}N_{n-1}(Tx,{\underline{\ell}}),\,N_{n-1}(Tx,{\underline{0}})+1].$

If ${\underline{\ell}}(n-1,Tx)={\underline{0}}$, then
$N_{n-1}(Tx,{\underline{0}})\geq\sup_{{\underline{\ell}}\not={\underline{0}}}N_{n-1}(Tx,{\underline{\ell}})$.
If ${\underline{\ell}}(n-1,Tx)\not={\underline{0}}$, then
$N_{n-1}(Tx,{\underline{0}})<\sup_{{\underline{\ell}}\not={\underline{0}}}N_{n-1}(Tx,{\underline{\ell}})$.
This shows (29).

######  Remark 2.4.

By (28), if $K_{n}$ is a uniform bound over $x$ of $N_{n}(x,{\underline{0}})$,
then $M_{n}(x)\leq K_{n}$. Likewise, if $N_{n}(x,{\underline{0}})\leq K_{n}$,
for a.e. $x$, then $M_{n}(x)\leq K_{n}$, for a.e. $x$.

Now we show that the set of $x\in X$ such that $\lim_{n}{M_{n}^{2}(x)\over
V_{n}(x)}=0$ has measure 0 or 1.

######  Lemma 2.5.

It holds: $\displaystyle\lim_{n}\,[{M_{n}^{2}(x)\over
V_{n}(x)}-{M_{n}^{2}(Tx)\over V_{n}(Tx)}]=0$. If $T$ is ergodic, there is a
constant $\gamma\in[0,1]$ such that
$\displaystyle\limsup_{n}{M_{n}^{2}(x)\over V_{n}(x)}=\gamma$ for a.e. $x$.

Proof. We use (31) and (29). Putting
$\varepsilon=1_{{\underline{\ell}}(n-1,Tx)={\underline{0}}}$, we have:

$\displaystyle|{M_{n}^{2}(x)\over V_{n}(x)}-{M_{n-1}^{2}(Tx)\over
V_{n-1}(Tx)}|=|{M_{n-1}^{2}(Tx)+\varepsilon(2M_{n-1}(Tx)+1)\over
V_{n-1}(Tx)+2N_{n-1}(Tx,{\underline{0}})+1}-{M_{n-1}^{2}(Tx)\over
V_{n-1}(Tx)}|$ $\displaystyle=|{\varepsilon(2M_{n-1}(Tx)+1)\over
V_{n}(x)}-{(2N_{n-1}(Tx,{\underline{0}})+1)\over
V_{n}(x)}\,{M_{n-1}^{2}(Tx)\over V_{n-1}(Tx)}|$
$\displaystyle\leq{2M_{n-1}(Tx)+1\over
V_{n}(x)}+{2N_{n-1}(Tx,{\underline{0}})+1\over V_{n}(x)}\leq{4M_{n-1}(Tx)\over
V_{n}(x)}+{2\over V_{n}(x)}\leq{4\over\sqrt{n}}+{2\over V_{n}(x)}.$

For the last inequality we use that either $M_{n}(x)\geq\sqrt{n}$, hence
${M_{n}(x)\over V_{n}(x)}\leq{1\over M_{n}(x)}\leq{1\over\sqrt{n}}$, or
$M_{n}(x)<\sqrt{n}$, hence ${M_{n}(x)\over V_{n}(x)}\leq{\sqrt{n}\over
n}={1\over\sqrt{n}}$. Therefore,

(33) $\displaystyle|{M_{n}^{2}(x)\over V_{n}(x)}-{M_{n-1}^{2}(Tx)\over
V_{n-1}(Tx)}|\to 0.$

Observe now that

$\displaystyle M_{n}(x)=M_{n-1}(x)+\varepsilon_{n},\text{ where
}\varepsilon_{n}=0\text{ or }=1,$

and $\varepsilon_{n}=1$ if and only if there is ${\underline{\ell}}_{n}$ such
that

$M_{n-1}(x)=\\#\\{1\leq k\leq n-1:\ f_{k}(x)={\underline{\ell}}_{n}\\}\text{
and }f_{n}(x)={\underline{\ell}}_{n}.$

We have

$M_{n}^{2}(x)=M_{n-1}^{2}(x)+c_{n},\text{ with
}c_{n}=\varepsilon_{n}(1+2M_{n-1}(x))$

and
$N_{n}(x,{\underline{\ell}})=N_{n-1}(x,{\underline{\ell}})+\varepsilon^{\prime}_{n}({\underline{\ell}})$,
with
$\varepsilon^{\prime}_{n}({\underline{\ell}})=1_{f_{n}(x)={\underline{\ell}}}$
and
$\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\,\varepsilon^{\prime}_{n}({\underline{\ell}})=1$.

Therefore,

$\displaystyle
V_{n}(x)=\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\,(N_{n-1}(x,{\underline{\ell}})+\varepsilon^{\prime}_{n}({\underline{\ell}}))^{2}=V_{n-1}(x)+2\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\,\varepsilon^{\prime}_{n}({\underline{\ell}})N_{n}(x,{\underline{\ell}}))+1,$
$\displaystyle 0\leq V_{n}(x)=V_{n-1}(x)+d_{n},\text{ with }d_{n}\leq
2M_{n}(x)+1.$ $\displaystyle|{M_{n}^{2}(x)\over V_{n}(x)}-{M_{n-1}^{2}(x)\over
V_{n-1}(x)}|=|{M_{n-1}^{2}(x)+c_{n}\over
V_{n-1}(x)+d_{n}}-{M_{n-1}^{2}(x)\over V_{n-1}(x)}|\leq{c_{n}\over
V_{n}(x)}+{d_{n}\over V_{n}(x)}\,{M_{n-1}^{2}(x)\over V_{n-1}(x)}$
$\displaystyle\leq{\varepsilon_{n}(1+2M_{n-1}(x)\over
V_{n}(x)}+{2M_{n}(x)+1\over V_{n}(x)}\,{M_{n-1}^{2}(x)\over
V_{n-1}(x)}\leq{2\over V_{n}(x)}+{4M_{n}(x)\over V_{n}(x)}\leq{2\over
V_{n}(x)}+{4\over\sqrt{n}}\to 0.$

From (33) and the convergence above, it follows $\lim_{n}\,[{M_{n}^{2}(x)\over
V_{n}(x)}-{M_{n}^{2}(Tx)\over V_{n}(Tx)}]=0$. By ergodicity of $T$, this shows
the lemma ∎

Case of a coboundary

The case when $f$ is coboundary degenerates. Indeed, the following holds:

######  Proposition 2.6.

If $f$ is a coboundary we have: a) $\liminf_{n}{M_{n}(x)\over n}>0$, for a.e.
$x$; b) there is a constant $\beta>0$ such that
${\frac{1}{n^{2}}}V_{n}(x)\to\beta$, for a.e. $x$; c) for a.e. $x$,
$\liminf_{n}{M_{n}^{2}(x)\over V_{n}(x)}>0$.

Proof. Suppose that $f$ is coboundary, $f=T\Phi-\Phi$. Since $f$ has values in
${\mathbb{Z}}^{d}$ and $T$ is ergodic, for all component $\Phi_{j}$ of $\Phi$,
$e^{2\pi i\Phi_{j}}$ is a constant. It follows that $\Phi$ has also its values
in ${\mathbb{Z}}^{d}$ up to an additive constant and we can assume that $\Phi$
has values in ${\mathbb{Z}}^{d}$.

a) We have $\liminf_{n}{M_{n}(x)\over
n}\geq\liminf_{n}\frac{1}{n}N_{n}(x,{\underline{0}})>0$, for a.e. $x$. The
positivity results the following simple argument:

For $R\geq 1$, let $A_{R}$ denote the set
$\cup_{{\underline{\ell}}:\|{\underline{\ell}}\|\leq
R}(\Phi={\underline{\ell}})$. Since, for each ${\underline{\ell}}$, by
Birkhoff’s theorem, $\lim_{n}\frac{1}{n}\sum_{0\leq k\leq
n-1}1_{\Phi(T^{k}x)={\underline{\ell}}}=\mu(\Phi={\underline{\ell}})$, it
holds

$\frac{1}{n}N_{n}(x,{\underline{0}})\geq\sum_{{\underline{\ell}}\in
A_{R}}\,1_{(\Phi={\underline{\ell}})}(x)\,\frac{1}{n}\sum_{k=0}^{n-1}1_{(\Phi={\underline{\ell}})}(T^{k}x)\to\sum_{{\underline{\ell}}\in
A_{R}}\,1_{(\Phi={\underline{\ell}})}(x)\,\mu(\Phi={\underline{\ell}}).$

Therefore, for every $R\geq 1$, $\liminf_{n}{M_{n}(x)\over
n}\geq\liminf_{n}{N_{n}(x,0)\over n}\geq\sum_{{\underline{\ell}}\in
A_{R}}\,1_{(\Phi={\underline{\ell}})}(x)\,\mu(\Phi={\underline{\ell}})$, and
the limit when $R\to\infty$ at right is $>0$, for a.e. $x$.

b) For $V_{n}$, we have:

$\displaystyle V_{n}(f,x)$
$\displaystyle=\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\,N_{n}^{2}(x,{\underline{\ell}})=\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\,\\#\\{0\leq
k\leq n-1:\ \Phi(T^{k}x)-\Phi(x)={\underline{\ell}}\\}^{2}$
$\displaystyle=\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\,\\#\\{0\leq k\leq
n-1:\
\Phi(T^{k}x)={\underline{\ell}}\\}^{2}=\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\,\bigl{(}\sum_{0\leq
k\leq n-1}1_{\Phi(T^{k}x)={\underline{\ell}}}\bigr{)}^{2},$ $\text{hence:
}{\frac{1}{n^{2}}}\sum_{{\underline{\ell}}\in A_{R}}\,\bigl{(}\sum_{0\leq
k\leq
n-1}1_{\Phi(T^{k}x)={\underline{\ell}}}\bigr{)}^{2}=\sum_{{\underline{\ell}}\in
A_{R}}\,\bigl{(}{\frac{1}{n}}\sum_{0\leq k\leq
n-1}1_{\Phi(T^{k}x)={\underline{\ell}}}\bigr{)}^{2}\to\sum_{{\underline{\ell}}\in
A_{R}}\,(\mu(\Phi={\underline{\ell}}))^{2}.$

This implies, for every $R\geq 1$,

$\liminf_{n}{\frac{1}{n^{2}}}\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\,\bigl{(}\sum_{0\leq
k\leq
n-1}1_{\Phi(T^{k}x)={\underline{\ell}}}\bigr{)}^{2}\geq\lim_{n}{\frac{1}{n^{2}}}\sum_{{\underline{\ell}}\in
A_{R}}\,\bigl{(}\sum_{0\leq k\leq
n-1}1_{\Phi(T^{k}x)={\underline{\ell}}}\bigr{)}^{2}=\sum_{{\underline{\ell}}\in
A_{R}}\,(\mu(\Phi={\underline{\ell}}))^{2}.$

It follows:
$\displaystyle\liminf_{n}{\frac{1}{n^{2}}}\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\,\bigl{(}\sum_{0\leq
k\leq
n-1}1_{\Phi(T^{k}x)={\underline{\ell}}}\bigr{)}^{2}\geq\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\,\mu(\Phi={\underline{\ell}})^{2}$.
For the complementary of $A_{R}$, it holds:

$\displaystyle\sum_{{\underline{\ell}}:\|{\underline{\ell}}\|>R}\bigl{(}\sum_{0\leq
k<n}1_{\Phi(T^{k}x)={\underline{\ell}}}\bigr{)}^{2}=\sum_{0\leq
j,k<n}\,\sum_{{\underline{\ell}}:\|{\underline{\ell}}\|>R}1_{\Phi(T^{j}x)={\underline{\ell}}}\,1_{\Phi(T^{k}x)={\underline{\ell}}}$
$\displaystyle\leq\sum_{0\leq
j,k<n}\,(\sum_{{\underline{\ell}}:\|{\underline{\ell}}\|>R}1_{\Phi(T^{j}x)={\underline{\ell}}})\,(\sum_{{\underline{\ell}}:\|{\underline{\ell}}\|>R}1_{\Phi(T^{k}x)={\underline{\ell}}})\leq\sum_{0\leq
j,k<n}1_{A_{R}^{c}(T^{j}x)}1_{A_{R}^{c}(T^{k}x)}=\bigl{(}\sum_{0\leq
k<n}1_{A_{R}^{c}(T^{k}x})\bigr{)}^{2}.$

It follows for the upper bound:

$\displaystyle\limsup_{n}{\frac{1}{n^{2}}}\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\,\bigl{(}\sum_{0\leq
k\leq n-1}1_{\Phi(T^{k}x)={\underline{\ell}}}\bigr{)}^{2}$
$\displaystyle\leq\lim_{n}\sum_{{\underline{\ell}}\in
A_{R}}\,\bigl{(}{\frac{1}{n}}\sum_{0\leq k\leq
n-1}1_{\Phi(T^{k}x)={\underline{\ell}}})^{2}+\lim_{n}\bigl{(}{\frac{1}{n}}\sum_{0\leq
k<n}1_{A_{R}^{c}(T^{k}x)}\bigr{)}^{2}$
$\displaystyle=\sum_{{\underline{\ell}}\in
A_{R}}\,(\mu(\Phi={\underline{\ell}}))^{2}+\mu(A_{R}^{c})^{2}\underset{R\to\infty}{\to}\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\,\mu(\Phi={\underline{\ell}})^{2}.$

This shows b) with
$\beta=\sum_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}\,\mu(\Phi={\underline{\ell}})^{2}>0$.

c) Follows from a) and b). ∎

######  Proposition 2.7.

There is a constant $\beta\geq 0$ such that, for a.e. $x$,
$\lim_{n}{V_{n}(x)\over n^{2}}=\beta$. We have $\beta>0$ if and only if the
cocycle $(T,f)$ is a coboundary.

Proof. The case of a coboundary follows from Proposition 2.6.

Suppose now that the cocycle is not a coboundary. From (26), we can write

$\displaystyle{V_{n}(x)\over n^{2}}$
$\displaystyle=\frac{1}{n}+\frac{2}{n}\sum_{k=1}^{n-1}\,\frac{1}{n}\sum_{j=0}^{n-k-1}(1_{f_{k}(T^{j}x)={\underline{0}}})$
$\displaystyle\leq\frac{1}{n}+\frac{2}{n}\sum_{k=1}^{n-1}\,\frac{1}{n}\sum_{j=0}^{n-1}(1_{f_{k}(T^{j}x)={\underline{0}}})=\frac{1}{n}+\frac{2}{n}\sum_{j=1}^{n-1}\,{N_{n}(T^{j}x,0)\over
n}.$

We will show that
$\displaystyle\frac{1}{n}\sum_{j=0}^{n-1}\,{N_{n}(T^{j}x,0)\over n}$ tend to 0
a.e.

By the ergodic theorem of Dunford and Schwarz (in the space of infinite
measure $X\times{\mathbb{Z}}$) applied to $\tilde{T}_{f}$ and $\phi_{0}={\bf
1}_{X\times\\{0\\}}$, which is bounded and in $L^{p}(X\times{\mathbb{Z}})$,
for every $p\geq 1$, we get a function $\tilde{\phi}_{0}(x)$ which is
$\tilde{T}_{f}$-invariant and in $L^{1}(X\times{\mathbb{Z}})$ and

$\lim_{n}{N_{n}(x,0)\over n}=\tilde{\phi}_{0}(x),\text{ a.s.}$

As $f$ is not a coboundary, $\tilde{\phi}_{0}$ is zero a.e. (cf. for instance
[12].)

Observe that $\|\sup_{n\geq L}{N_{n}(x,0)\over n}\|_{2}\to 0$, as $L$ goes to
$+\infty$. Indeed, for every $0<\varepsilon\leq 1$, letting
$A_{\varepsilon,L}:=\\{x:\sup_{n\geq L}{N_{n}(x,0)\over n}>\varepsilon\\}$, we
have $\mu(A_{\varepsilon,L})\to 0$, when $L\to+\infty$. Since
${N_{n}(x,0)\over n}\leq 1$, it follows, for $L$ big enough:

$\int\bigl{(}\sup_{n\geq L}({N_{n}(x,0)\over
n})\bigr{)}^{2}\,d\mu\leq\varepsilon^{2}+\mu(A_{\varepsilon,L})\leq
2\varepsilon.$

We put $\displaystyle\Lambda_{n}(x):=\sup_{s\geq n}{N_{s}(x,0)\over s}$. By
the previous observation, we have $\lim_{n}\|\Lambda_{n}\|_{2}=0$.

Let us consider the following maximal function for the action of $T$:

(34) $\displaystyle\tilde{\Lambda}_{n}(x)=\sup_{1\leq
r<\infty}\frac{1}{r}\sum_{j=0}^{r-1}\Lambda_{n}(T^{j}x)=\sup_{1\leq
r<\infty}\frac{1}{r}\sum_{j=0}^{r-1}\sup_{s\geq n}{N_{s}(T^{j}x,0)\over s}.$

From a classical maximal inequality, we have $\|\tilde{\Lambda}_{n}\|_{2}\leq
2\|\Lambda_{n}\|_{2}\to 0$.

Observe also that, from the definition of $\tilde{\Lambda}_{n}$ in (34), the
following inequalities hold:

$\tilde{\Lambda}_{n}(x)\geq\sup_{r,s\geq
n}\frac{1}{r}\sum_{j=0}^{r-1}\,{N_{s}(T^{j}x,0)\over
s}\geq\frac{1}{n}\sum_{j=0}^{n-1}\,{N_{n}(T^{j}x,0)\over n}.$

The sequence $\displaystyle\sup_{r,s\geq
n}\frac{1}{r}\sum_{j=0}^{r-1}\,{N_{s}(T^{j}x,0)\over s}$ is non negative and
decreasing. Since $\|\tilde{\Lambda}_{n}\|_{2}\to 0$, the $L_{2}$-norm of its
limit in $(X,\mu)$ is zero. The result follows. ∎

######  Remark 2.8.

(see also section 4 and [25])

Let $(U_{\underline{\ell}})_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}$ be a r.f.
of square integrable r.v.’s on a probability space
$(\Omega,\mathcal{F},{\mathbb{P}})$ stationary in the weak sense and such that
$\sum_{\underline{\ell}}|\langle
U_{\underline{\ell}},U_{{\underline{0}}}\rangle|<+\infty$. By (4) and
Proposition 2.7, if $f$ is not a coboundary, it holds

$\frac{1}{n^{2}}\,\|\sum_{k=0}^{n-1}U_{f_{k}(x)}\|_{2}^{2}\,\leq
C\,{V_{n}(x)\over n^{2}}\to 0,\text{ for }\mu\text{-a.e. }x.$

Another result of norm convergence whose proof is like the proof of
Proposition 1.2 is the following. Suppose that the r.f. is stationary. Let
$\varphi$ be an observable on the dynamical system
$(\Omega,{\mathbb{P}},\theta)$ with a spectral measure $\nu_{\varphi}$. We
have:

$\int_{\Omega}|\sum_{j=0}^{n-1}\varphi\circ\theta^{z_{j}}|^{2}\,d{\mathbb{P}}=\int_{{\mathbb{T}}^{1}}|\sum_{j=0}^{n-1}e^{2\pi
iz_{j}t}|^{2}\,d\nu_{\varphi}(t).$

Assume that $\nu_{\varphi}$ is absolutely continuous with respect to the
Lebesgue measure on the torus, and let $\rho\in L^{1}(d{\underline{t}})$ such
that $d\nu_{\varphi}({\underline{t}})=\rho({\underline{t}})d{\underline{t}}$.
For $\varepsilon>0$ there is $M$ such that
$\int_{\rho>M}\,\rho\,d{\underline{t}}<\varepsilon$. We have

$\displaystyle\frac{1}{n^{2}}\,\int_{{\mathbb{T}}^{d}}|\sum_{j=0}^{n-1}e^{2\pi
i\langle z_{j},{\underline{t}}\rangle}|^{2}\,d\nu_{\varphi}({\underline{t}})$
$\displaystyle\leq$
$\displaystyle\frac{M}{n^{2}}\,\int_{{\mathbb{T}}^{d}}|\sum_{j=0}^{n-1}e^{2\pi
i\langle
z_{j},{\underline{t}}\rangle}|^{2}\,d{\underline{t}}+\int_{\rho>M}\,\rho\,d{\underline{t}}\leq
M{V_{n}\over n^{2}}+\varepsilon.$

This shows that $\displaystyle{V_{n}\over n^{2}}\to 0$ implies
$\displaystyle\frac{1}{n^{2}}\,\int_{\Omega}|\sum_{j=0}^{n-1}\varphi\circ\theta^{z_{j}}|^{2}\,d{\mathbb{P}}\to
0$. This is satisfied by every $\varphi\in L^{2}({\mathbb{P}})$, if the
dynamical system has a Lebesgue spectrum.

In particular, taking $z_{k}=f_{k}(x)$, by Proposition 2.7, if $f$ is not a
coboundary, it holds

$\frac{1}{n^{2}}\,\int_{\Omega}|\sum_{j=0}^{n-1}\varphi(\theta^{f_{j}(x)}\omega)|^{2}\,d{\mathbb{P}}(\omega)\to
0,\text{ for a.e. }x.$

When the spectral density is square integrable, as we have seen in Proposition
1.2, the pointwise convergence holds under quantitative hypothesis on the
sequence $(z_{k})$.

###  2.2. Non centered cocycles

In an ergodic dynamical system $(X,\mu,T)$, if $f:X\to{\mathbb{R}}$ is an
integrable function with $\mu(f)>0$, by the ergodic theorem for the ergodic
sums $S_{n}^{T}f(x)=\sum_{k=0}^{n-1}f(T^{k}x)$, it holds for a.e. $x$:
$\lim_{n}\frac{1}{n}S_{n}f(x)>0$ and therefore
$\lim_{n}S_{n}^{T}f(x)=+\infty$. If $f$ has values in ${\mathbb{Z}}$, as the
process $S_{n}^{T}f(x)$ visits finitely often each site, one can think there
is a chance that the following condition is satisfied:

(35) $\displaystyle\lim_{n}{M^{2}_{n}(T,f,x)\over V_{n}(T,f,x)}=0.$

A case where (35) is satisfied is the following: let $X$ be a topological
compact space, $T:X\to X$ a continuous map, which is uniquely ergodic with
$\mu$ as unique invariant measure. Let $f:X\to{\mathbb{Z}}$ be an integrable
function such that $\mu(f)\not=0$. Assume $f$ to be Riemann-integrable (i.e.
such that, for every $\varepsilon>0$, there are two continuous functions
$\psi_{0},\psi_{1}$ with $\psi_{0}\leq f\leq\psi_{1}$ and
$\mu(\psi_{1}-\psi_{0})\leq\varepsilon$).

Then, the ergodic means of $f$ converge uniformly, and this implies the
existence of $N$ such that
$\frac{1}{n}|S_{n}^{T}f(x)|\geq\frac{1}{2}|\mu(f)|>0$ for $n\geq N$ and every
$x$. It follows that the number of visits of $S_{n}^{T}f(x)$ to 0 is $\leq N$,
for every $x$. By remark 2.4, $M_{n}(x)\leq N$, for every $x$, and a fortiori
(35) is satisfied.

Nevertheless, we will see that $(\ref{condi00x})$ may fail in non uniform
cases: there are dynamical systems and sets $B$ of positive measure such that,
for $f=1_{B}$,

(36) $\displaystyle\limsup_{n}{M^{2}_{n}(T,f,x)\over V_{n}(T,f,x)}=1.$

###  2.3. Counterexamples

In this subsection, we construct a transient counterexample, and also a
recurrent counterexample with a function $f$ of null integral such that (36)
is satisfied.

To construct these counterexamples, we start by considering a general ergodic
dynamical system $(X,\mu,T)$ and a measurable set $B\subset X$ of positive
measure. Let $T_{B}$ be the induced map on $B$, $R(x)=R^{B}(x)=\inf\\{k\geq
1:T^{k}x\in B\\}$ the first return time of $x$ in $B$ and
$R_{n}(x)=R_{n}^{B}(x):=\sum_{k=0}^{n-1}R(T_{B}^{k}x)$ the $n$-th return time
of $x$ in $B$.

We take $x\in B$. If $f$ is a function such that $f=0$ outside $B$, the
position of the sums up to time $R_{n-1}(x)$ are the positions of the ergodic
sums $S_{n}^{T_{B}}f$ for the induced map up to time $n$, that is:

$\\{f(x),f(x)+f(T_{B}x),...,f(x)+f(T_{B}x)+...+f(T_{B}^{n-1}x)\\}.$

For a site $\ell$, the number of visits up to time $R_{n-1}(x)$ of the ergodic
sums for $T$ is

$N_{R_{n-1}(x)}(x,\ell)=\sum_{k=0}^{n-1}R^{B}(T_{B}^{k}x)\,1_{S_{k}^{T_{B}}f(x)=\ell}$

and therefore

(37) $\displaystyle
V_{R_{n-1}(x)}(T,x)=\sum_{\ell}[\sum_{k=0}^{n-1}R^{B}(T_{B}^{k}x)\,1_{S_{k}^{T_{B}}f(x)=\ell}]^{2}.$

Case $f=1_{B}$. Clearly $\sum_{k=0}^{n-1}f(T_{B}^{k}x)=n$. For the map $T$,
the ergodic sums of $f$ are incremented by 1 when and only when the iterates
$T^{j}x$ visit the set B. Otherwise, they stay fixed. The times of visits in
$B$, for $x\in B$, are $0,R(x),R(x)+R(T_{B}x),...$. We have:

$\displaystyle\text{for }x\in B,\sum_{j=0}^{R_{n-1}(x)+t}f(T^{j}x)=n,\text{
for }t=0,...,R_{n}(x)-R_{n-1}(x)-1.$

For $N_{n}(T,x,\ell)=N_{n}(T,f,x,\ell)$, it holds:

$\displaystyle N_{n}(T,x,\ell)$ $\displaystyle=$ $\displaystyle 0,\text{ if
}n<R_{\ell}(x),$ $\displaystyle=$ $\displaystyle t,\text{ if
}n=R_{\ell}(x)+t,\text{ with }0\leq t<R_{\ell+1}(x)-R_{\ell}(x),$
$\displaystyle=$ $\displaystyle
R_{\ell+1}(x)-R_{\ell}(x)=R(T_{B}^{\ell}x),\text{ if }n\geq R_{\ell+1}(x).$

For $L\geq 1$, we have for the time preceding the $L$-th return to the basis
for $f=1_{B}$:

(38) $\displaystyle M_{R_{L}(x)-1}(T,f,x)=\max_{\ell\leq L}R(T_{B}^{\ell}x),\
V_{R_{L}(x)-1}(T,f,x)=\sum_{\ell\leq L}R^{2}(T_{B}^{\ell}x).$

In order to compute an explicit example, it is easier to start from a given
map $S$ and construct a special flow $T$ over this map.

Let $\varphi:X\to{\mathbb{N}}$ be integrable and $\geq 1$. The (discrete time)
special map $T=T_{\varphi}$ is defined

$\displaystyle\text{ on }\tilde{X}:=\\{(x,k),\ x\in
X,k=0,...,\varphi(x)-1\\}\subset X\times{\mathbb{R}},$ $\displaystyle\text{ by
}T(x,k):=(x,k+1),\text{ if }0\leq k<\varphi(x)-1,\ :=(Sx,0),\text{ if
}k=\varphi(x)-1.$

Let $\tilde{\mu}$ be the probability measure defined on $\tilde{X}$ by
$\tilde{\mu}(A\times\\{k\\})=\mu(\varphi)^{-1}\,\mu(A)$, for $k\geq 0$ and
$A\subset\\{x:k\leq\varphi(x)-1\\}$. It is $T_{\varphi}$-invariant. The space
$X$ can be identified with the subset $B=\\{(x,0),x\in X\\}$ of $\tilde{X}$
with normalized measure. The set $B$ is the basis and $\varphi-1$ the roof
function of the special map $T_{\varphi}$.

As for the map $S$ we will take an ergodic rotation, the special flow
$T_{\varphi}$ will be also ergodic for the measure $\tilde{\mu}$ on
$\tilde{X}$.

Observe that the recurrence time $R(x)=R^{B}(x)$ for the special flow in the
basis $B$ is $\varphi(x)$ and the $n$-th return time of $x$ in $B$ is
$R_{n}(x)=R_{n}^{B}(x)=\sum_{k=0}^{n-1}\varphi(S^{k}x)$.

For $S$, let us take a rotation $S=S_{\alpha}$ on
$X={\mathbb{T}}/{\mathbb{Z}}$ by $\alpha{\rm\ mod\,}1$, where $\alpha$ is
irrational. We denote by $q_{n}$ the denominators of $\alpha$. We will
construct the measure preserving transformation which is the special flow
(with discrete time) over $S_{\alpha}$ with a roof function $\varphi$ such
that, for cocycle generated by $1_{B}$ in the system
$(\tilde{X},\tilde{\mu},T)$, $\displaystyle\liminf_{n}{V_{n}(T,x)\over
M_{n}^{2}(T,x)}=1$.

We will use the next lemma with $p=p_{n},q=q_{n}$, the numerators and
denominators of $\alpha$.

######  Lemma 2.9.

Let $p,q\geq 1$, $(p,q)=1$, be such that $|\alpha-p/q|<1/q^{2}$. For every
$x$, there is a value $0\leq i<q$ such that $x+i\alpha{\rm\
mod\,}1\in[0,2/q]$.

More generally, for every interval $I$ of length $2/q$, for every $x$, there
is a value $0\leq i<q$ such that $x+i\alpha{\rm\ mod\,}1\in I$.

Proof. It is well known that there is exactly one value of $j\alpha\text{ mod
}1$, for $0\leq j<q$, in each interval $[{\ell\over q},{\ell+1\over q}[$,
$\ell=0,...,q-1$. Let us recall a proof. For $j=0$, $j\alpha\in[0,1/q[$. The
map $j\to\ell_{j}=jp{\rm\ mod\,}q$, which is injective, is a permutation of
the set $\\{1,...,q-1\\}$ onto itself. We have $\alpha=p/q+\gamma$, with
$|\gamma|<1/q^{2}$.

Assuming $\gamma>0$, it follows: $j\alpha{\rm\ mod\,}1\in[{\ell_{j}\over
q},{\ell_{j}\over q}+{j\over q^{2}}]\subset[{\ell_{j}\over q},{\ell_{j}+1\over
q}[,\text{ for }j=1,...,q-1$. The case $\gamma<0$ is treated the same way.

Now let us prove the first point. Let $x$ be in $[0,1[$. There is
$i_{0}\in\\{0,...,q-1\\}$ such that $x={i_{0}\over q}+\theta$, with
$0\leq\theta<1/q$. By the claim, there is $i\in[0,q[$ such that $i\alpha{\rm\
mod\,}1\in[{q-i_{0}\over q},{q-i_{0}+1\over q}]$. Hence $x+i\alpha{\rm\
mod\,}1\in[\theta,\frac{1}{q}+\theta]\subset[0,\frac{2}{q}]$. ∎

Let $(\lambda_{n})$ be an increasing sequence of positive integers which will
be subjected below to growth conditions. First we assume that it satisfies the
condition:

(39) $\displaystyle q_{\lambda_{n+1}}\geq 3q_{\lambda_{n}},\forall n\geq 1.$

Denote by $J_{n}$ the interval $\displaystyle J_{n}=[{3\over
q_{\lambda_{n+1}}},{3\over q_{\lambda_{n}}}]$. For the roof function, we take,
with $\varepsilon_{n}=\frac{1}{n^{2}}$,

$\displaystyle\varphi=1+\sum_{n\geq
1}\lfloor\varepsilon_{n}q_{\lambda_{n}}\rfloor 1_{J_{n}}.$

The function $\varphi$ is integrable: $\int\varphi d\mu\leq
1+3\sum_{n}\varepsilon_{n}$. Observe also that, by (39), the length of $J_{n}$
is $>2/q_{\lambda_{n}}$ and that $(\varepsilon_{n}q_{\lambda_{n}})$ is not
decreasing for $n\geq 2$ .

Let $x$ be in the basis. By construction, the orbit of $x$ under the iteration
of $T_{\varphi}$ is that of the rotation $S_{\alpha}$ until it enters the set
$B^{c}$, complementary of $B$ at some time. Then it stays in this set, until
it reaches the roof and comes down to the basis. Then the dynamic is that of
the rotation, until again $S_{\alpha}^{j}x$ falls in the set $\varphi>1$ and
so on.

Let $W_{n}(x)$ be the first visit of $S^{j}x$ in $J_{n}$. By lemma 2.9, we
have $W_{n}(x)\leq q_{\lambda_{n}}$.

Now we choose $f$ to get a transient counterexample and a recurrent one.

Transient counterexample.

We take $f=1$ on the basis and 0 outside.

The sequence $(\lambda_{n})$ is taken such that

(40) $\displaystyle q_{\lambda_{n}}\geq n\,(q_{\lambda_{n-1}})^{2},n\geq 1.$

By (38), we obtain (recall that now $T_{B}$, the induced map in the basis $B$,
is the rotation $S=S_{\alpha}$ and $R(T_{B}^{j}x)=\varphi(S_{\alpha}^{j}x)$):

(41) $\displaystyle M_{R_{W_{n}(x)}(x)-1}(T,x)$ $\displaystyle=$
$\displaystyle\max_{j\leq W_{n}(x)}\varphi(S^{j}x),$ (42) $\displaystyle
V_{R_{W_{n}(x)}(x)-1}(T,x)$ $\displaystyle=$ $\displaystyle\sum_{j\leq
W_{n}(x)}\varphi^{2}(S^{j}x).$

In the above formula, $\varphi(S^{j}x)$ is either 1 or (for some $k\leq n-1$)
$1+\lfloor\varepsilon_{k}q_{\lambda_{k}}\rfloor\leq
1+\varepsilon_{n-1}q_{\lambda_{n-1}}$, excepted for the last term which is
$1+\lfloor\varepsilon_{n}q_{\lambda_{n}}\rfloor$.

The maximum in (41) (given by the first visit to $J_{n}$) is
$1+\lfloor\varepsilon_{n}q_{\lambda_{n}}\rfloor\geq\varepsilon_{n}q_{\lambda_{n}}$.
As we have seen, this first visit for the iterates $S^{j}x$ occurs at a time
$\leq q_{\lambda_{n}}$. It follows by (40):

$\displaystyle{V_{R_{W_{n}(x)}(x)-1}(T,x)\over
M^{2}_{R_{W_{n}(x)}(x)-1}(T,x)}$ $\displaystyle\leq$ $\displaystyle
q_{\lambda_{n}}{(\varepsilon_{n-1}\,q_{\lambda_{n-1}})^{2}\over(\varepsilon_{n}\,q_{\lambda_{n}}-1)^{2}}+1\leq({\varepsilon_{n-1}\over\varepsilon_{n}})^{2}\,{(q_{\lambda_{n-1}})^{2}\over
q_{\lambda_{n}}}{1\over(1-(\varepsilon_{n}\,q_{\lambda_{n}})^{-1})^{2}}+1$
$\displaystyle\leq$ $\displaystyle 2\,({n\over
n-1})^{2}\,{(q_{\lambda_{n-1}})^{2}\over
q_{\lambda_{n}}}+1\leq\frac{4}{n}+1,\text{ for }n\text{ big enough}.$

This shows: $\displaystyle\limsup_{n}{M^{2}_{n}(T,f,x)\over V_{n}(T,f,x)}=1$.
The result is proved for $x$ in the basis $B$, but is satisfied for a.e.
$x\in\tilde{X}$, since $\displaystyle\limsup_{n}{M^{2}_{n}(T,f,x)\over
V_{n}(T,f,x)}$ is a.e. constant by ergodicity of the special flow and Lemma
2.5.

Remark that $S_{k}f(x)\to+\infty$ for every point $x$.The sequence
$(N_{n}(x,0))$ is bounded for every $x$, but not uniformly in $x$.

Recurrent counterexample.

In order to obtain a recurrent counterexample, we now use a special cocycle
over a rotation by $\alpha$ (with $\alpha$ bpq) studied later (see Subsection
3.3).

Let $f$ defined on the basis by
$f(x)=1_{[0,\frac{1}{2}[}(x)-1_{[\frac{1}{2},1[}(x)$ and 0 outside, and
$S_{k}f(x)=\sum_{i=0}^{k-1}f(x+i\alpha{\rm\ mod\,}1)$. By (37), we have

$\displaystyle V_{R_{n-1}(x)}(T,f,x)$ $\displaystyle=$
$\displaystyle\sum_{\ell}[\sum_{k=0}^{n-1}\varphi(x+k\alpha)\,1_{S_{k}f(x)=\ell}]^{2}$
$\displaystyle=$
$\displaystyle\sum_{\ell}[\sum_{k=0}^{n-1}(1+\sum_{j}\varepsilon_{j}q_{\lambda_{j}}1_{J_{j}}(x+k\alpha))\,1_{S_{k}f(x)=\ell}]^{2}.$

Observe that for a constant $C$,
$1+\sum_{j<n}\varepsilon_{j}q_{\lambda_{j}}1_{J_{j}}(x+k\alpha))\leq
Cq_{\lambda_{n-1}}$. Using the bounds for the special function $f$ and
$\alpha$ bpq, this implies:

$\displaystyle V_{R_{W_{n-1}(x)}}(T,f,x)$ $\displaystyle\leq$
$\displaystyle\sum_{\ell}[\sum_{k=0}^{q_{\lambda_{n}}}(1+\sum_{j<n}\varepsilon_{j}q_{\lambda_{j}}1_{J_{j}}(x+k\alpha))\,1_{S_{k}f(x)=\ell}]^{2}$
$\displaystyle\leq$ $\displaystyle
C^{2}\sum_{\ell}[\sum_{k=0}^{q_{\lambda_{n}}}q_{\lambda_{n-1}}\,1_{S_{k}f(x)=\ell}]^{2}$
$\displaystyle\leq$ $\displaystyle
C^{2}q_{\lambda_{n-1}}^{2}\,\sum_{\ell}[\sum_{k=0}^{q_{\lambda_{n}}}\,1_{S_{k}f(x)=\ell}]^{2}\leq
C^{2}q_{\lambda_{n-1}}^{2}\,q_{\lambda_{n}}^{2}/\sqrt{\log q_{\lambda_{n}}}.$

Put $L_{n}=S_{W_{n}(x)}f(x)$ for the site visited by the cocycle when $S^{j}x$
enters $J_{n}$. We have

$M_{R_{W_{n}(x)}}(T,f,x)\geq
N_{R_{W_{n}(x)}}(T,f,x,L_{n}(x))=\varepsilon_{n}q_{\lambda_{n}}.$

Hence:

$\displaystyle 0\leq{V_{R_{W_{n}(x)}}(T,f,x)\over
M^{2}_{R_{W_{n}(x)}}(T,f,x)}-1$ $\displaystyle\leq$ $\displaystyle
C^{2}{q_{\lambda_{n-1}}^{2}\,q_{\lambda_{n}}^{2}\over\sqrt{\log
q_{\lambda_{n}}}}{1\over\varepsilon_{n}^{2}q_{\lambda_{n}}^{2}}=C^{2}{n^{4}q_{\lambda_{n-1}}^{2}\over\sqrt{\log
q_{\lambda_{n}}}}.$

Now, we choose a growth condition on $(\lambda_{n})$ stronger than (40), such
that the above bound tends to 0.

This shows the result for $x$ in the basis, hence on the whole space using
again Lemma 2.5.

##  3\. Examples

In general, for a dynamical system $(X,\mu,T)$ and a cocycle $(T,f)$, it seems
difficult to get a precise estimate of the quantities
$N_{n}(x,{\underline{\ell}}),M_{n}(x),V_{n}(x)$. In this section we present
two types of cocycles for which this is possible, first in the case of strong
stochastic properties, in particular for the classical case of random walks,
then when they are generated by step functions over rotations.

###  3.1. Random walks

1-dimensional cocycle satisfying the LIL.

We start be a remark on the the law of iterated logarithm (LIL). Suppose that
$(T,f)$ is a 1-dimensional cocycle which satisfies the LIL. Then for a
constant $c_{1}>0$, for a.e. $x$, the inequality $|f_{n}(x)|>c_{1}\,(n\
\ln\ln\,n)^{\frac{1}{2}}$ is satisfied only for finitely many values of $n$.
This implies that, for a.e. $x$, there is $N(x)$ such that
$|f_{n}(x)|\leq(c_{1}\,n\ \ln\ln\,n)^{\frac{1}{2}}$, for $n\geq N(x)$; so
that, for $N(x)\leq k<n$, $|f_{k}(x)|\leq(c_{1}\,k\
\ln\ln\,k)^{\frac{1}{2}}\leq(c_{1}\,n\ \ln\ln\,n)^{\frac{1}{2}}$.

Therefore we have ${\rm Card}(\mathcal{R}_{n}(x))\leq c_{2}(x)\,(n\
\ln\ln\,n)^{\frac{1}{2}}$, with an a.e. finite constant $c_{2}(x)$.

In dimension 1, by (25), we get that for a.e. $x$ there is $c(x)>0$ such that

$\displaystyle V_{n}(x)\geq
C(x)\,n^{\frac{3}{2}}\,(\ln\ln\,n)^{-\frac{1}{2}}.$

The case where a LIL is valid includes the case of a 1-dimensional r.w.
centered with finite variance, but also the class of cocycles for which a
martingale method can be used.

Random walks.

Now we consider sequences given by a random walk. For random walks in
${\mathbb{Z}}^{d}$, the quantities $V_{n}(x),M_{n}(x)$ have been studied in
many papers since the 50’s. $M_{n}(x)$ is called “maximal multiplicity of
points on a random walk” by Erdös and Taylor [16]. Below, we give a brief
survey of several results for r.w.s. First we recall some definitions.

Let $(\zeta_{i})_{i\geq 0}$ be a sequence of i.i.d. random vectors on a
probability space $(X,\,\mu)$ with values in ${\mathbb{Z}}^{d}$ and common
probability distribution $\nu$. The associated random walk (r.w.) $Z=(Z_{n})$
in ${\mathbb{Z}}^{d}$ starting from ${\underline{0}}$ is defined by
$Z_{0}:={\underline{0}}$,

$Z_{n}:=\zeta_{0}+...+\zeta_{n-1},n\geq 1.$

A r.w. can be seen as a special case of cocycle. Indeed, the r.v.’s
$\zeta_{i}$ can be viewed as the coordinate maps on $(X,\,\mu)$ obtained as
$({\mathbb{Z}}^{d})^{\mathbb{Z}}$ equipped with the product measure
$\nu^{\otimes{\mathbb{Z}}}$ and with the shift $T$ acting on the coordinates.
We have $\zeta_{i}=\zeta_{0}\circ T^{i}$ and the cocycle relation
$Z_{n+n^{\prime}}=Z_{n}+Z_{n^{\prime}}\circ T^{n},\forall n,n^{\prime}\geq 0$.

Let $\cal
S:=\\{{\underline{\ell}}\in{\mathbb{Z}}^{d}:{\mathbb{P}}(\zeta_{0}={\underline{\ell}})>0\\}$
be the support of $\nu$ and $L$ the sub-lattice of ${\mathbb{Z}}^{d}$
generated by $\mathcal{S}$. Let $D$ be the sub-lattice of ${\mathbb{Z}}^{d}$
generated by
$\\{{\underline{\ell}}-{\underline{\ell}}^{\prime},{\underline{\ell}},{\underline{\ell}}^{\prime}\in\cal
S\\}$.

For simplicity (and without loss of generality) in what follows we will assume
that the random walk $Z$ is aperiodic ($L={\mathbb{Z}}^{d})$. We exclude also
the “deterministic” case (i.e., when
${\mathbb{P}}(\zeta_{0}={\underline{\ell}})=1$ for some
${\underline{\ell}}\in{\mathbb{Z}}^{d}$) in dimension 1 (the deterministic
case in higher dimension is excluded by aperiodicity).

Notice that all the pointwise limits or bounds mentioned now for random walks
are a.s. statements. These bounds will show that conditions (22), (21) are
satisfied by $V_{n}(x),M_{n}(x)$ a.s. for on random walks under mild
assumptions.

Recurrence/transience.

Recall that a r.w. $Z=(Z_{n})$ is recurrent if
$\sum_{n=1}^{\infty}\mu(Z_{n}={\underline{0}})=+\infty$ and otherwise
transient.

Recurrence occurs if and only if $\mu(Z_{n}={\underline{0}}\text{ infinitely
often})=1$, and transience if and only if $\mu(Z_{n}={\underline{0}}\text{
infinitely often})=0$ (cf. [8], [9]).

For an aperiodic r.w. $Z$ in dimension $d$ with a moment of order 2 (for
$d=1$, a moment of order 1 suffices), for $d=1,2$, $Z$ is recurrent if and
only if it is centered. For $d\geq 3$, it is always transient.

Variance.

Let $(X_{\underline{\ell}},{\underline{\ell}}\in{\mathbb{Z}}^{d})$ be a
stationary centered r.f. with summable correlation and spectral density
$\rho$. We have

$\frac{1}{n}\|\sum_{k=1}^{n-1}X_{Z_{k}(x)}\|_{2}^{2}=\int_{{\mathbb{T}}^{d}}\frac{1}{n}|\sum_{k=0}^{n-1}e^{2\pi
i\langle
Z_{k}(x),{\underline{t}}\rangle}|^{2}\,d{\underline{t}}=\int_{{\mathbb{T}}^{d}}\frac{1}{n}K_{n}(x,{\underline{t}})\,\rho({\underline{t}})\,d{\underline{t}},$

where, using (17) with $z_{k}=Z_{k}(x)$ and $Z_{k}(x)-Z_{j}(x)=Z_{k}(T^{j}x)$,
$\frac{1}{n}K_{n}$ reads

(43) $\displaystyle\frac{1}{n}K_{n}(x,{\underline{t}})$
$\displaystyle=1+2\sum_{\underline{\ell}}\bigl{(}\sum_{k=1}^{n-1}\,\frac{1}{n}\sum_{j=0}^{n-k-1}1_{Z_{k}(T^{j}x)={\underline{\ell}}}\bigr{)}\,e^{2\pi
i\langle{\underline{\ell}},t\rangle}.$

As already recalled, the existence of the asymptotic variance
$\lim_{n}V_{n}(x)^{-1}\int|\sum_{k=0}^{n-1}X_{Z_{k}(x)}|^{2}d\mu$ has been
shown in [11] and the positivity of the limit has been discussed.

The asymptotic variance may be zero in case a coboundary condition is
satisfied. An interesting situation is that of the sums along a transient (non
deterministic) r.w., where the asymptotic variance is always $>0$. Below we
will recall briefly a proof.

Transient case

For a transient random walk we use the following general result (Lemma 3.14 in
[11]):

######  Lemma 3.1.

If $(X,\mu,T)$ is an ergodic dynamical system and $(\varphi_{k})_{k\geq 1}$ a
sequence of functions in $L^{1}(X,\mu)$ such that $\sum_{k\geq
1}\|\varphi_{k}\|_{1}<\infty$, then

(44)
$\displaystyle\lim_{n}\frac{1}{n}\sum_{k=1}^{n-1}\sum_{j=0}^{n-k-1}\varphi_{k}(T^{j}x)=\sum_{k=1}^{\infty}\int\varphi_{k}\,d\mu,\text{
for a.e. }x.$

Therefore, for a transient random walk, we obtain for $\mu$-a.e. $x$:

$\lim_{n}{V_{n}(x)\over
n}=1+2\lim_{n}\sum_{k=1}^{n-1}\,\frac{1}{n}\sum_{j=0}^{n-k-1}1_{Z_{k}(T^{j}x)={\underline{0}}}=1+2\sum_{k=1}^{\infty}\mu(Z_{k}={\underline{0}})<+\infty$

and the normalisation for the variance is by $n$ up to a finite constant
factor.

Variance in the non deterministic transient case.

Now we recall the proof of the positivity of the asymptotic variance.

Let $\Psi({\underline{t}})={\mathbb{E}}[e^{2\pi
i\langle\zeta_{0},{\underline{t}}\rangle}],\,{\underline{t}}\in{\mathbb{T}}^{d}$.
Observe that $\Psi({\underline{t}})\not=1$ for
${\underline{t}}\not={\underline{0}}$ in ${\mathbb{T}}^{d}$, when the r.w. is
aperiodic and $|\Psi({\underline{t}})|<1$, for
${\underline{t}}\not\in\Gamma_{1}$, where $\Gamma_{1}$ is the closed subgroup
$\\{{\underline{t}}\in{\mathbb{T}}^{d}:e^{2\pi
i\langle{\underline{r}},{\underline{t}}\rangle}=1,\forall{\underline{r}}\ \in
D\\}$. We put, for
${\underline{t}}\in{\mathbb{T}}^{d}{\setminus\\{{\underline{0}}\\}}$ and
$0\leq\lambda<1$,

$\displaystyle\Phi({\underline{t}}):={1-|\Psi({\underline{t}})|^{2}\over|1-\Psi({\underline{t}})|^{2}}=\Re
e[{1+\Psi({\underline{t}})\over 1-\Psi({\underline{t}})}],$
$\displaystyle\Phi_{\lambda}({\underline{t}}):={1-\lambda^{2}|\Psi({\underline{t}})|^{2}\over|1-\lambda\Psi({\underline{t}})|^{2}}=-1+2\sum_{k=0}^{\infty}\lambda^{k}\Re
e(\Psi({\underline{t}})^{k})=-1+2\sum_{k=0}^{\infty}\lambda^{k}\mu(Z_{k}={\underline{\ell}})\,\cos(2\pi\langle{\underline{\ell}},{\underline{t}}\rangle),$

where the last relation follows from $\Re e(\Psi({\underline{t}})^{k})=\Re
e({\mathbb{E}}[e^{2\pi i\langle
Z_{k},{\underline{t}}\rangle}])=\sum_{\underline{\ell}}\mu(Z_{k}={\underline{\ell}})\,\cos(2\pi\langle{\underline{\ell}},{\underline{t}}\rangle)$.

We put $\Phi({\underline{0}})=0$.The function $\Phi$ is even, non-negative and
$\Phi({\underline{t}})=0$ only on $\Gamma_{1}$, which is
$\not={\mathbb{T}}^{d}$ when the r.w. is non deterministic (if the r.w. is
deterministic, $\mu(\zeta_{0}={\underline{\ell}})=1$ for some
${\underline{\ell}}\in{\mathbb{Z}}^{d}$ and this implies
$|\Psi({\underline{t}})|\equiv 1$, but this case is excluded). Therefore
$\Phi$ is $\not=0$ a.e. for the Lebesgue measure on ${\mathbb{T}}^{d}$.

######  Proposition 3.2.

(cf. [28]) Let $Z=(Z_{n})$ be a transient aperiodic random walk in
${\mathbb{Z}}^{d}$. There is a non-negative constant $M$ such that the Fourier
coefficients of $\frac{1}{n}K_{n}$ converges to those of
$\Phi+M\delta_{\underline{0}}$ and
$\displaystyle\lim_{n}\int\frac{1}{n}K_{n}\,\rho\,d{\underline{t}}>0$.

Proof. We use that, if $(Z_{n})$ is a transient, for all
${\underline{\ell}}\in{\mathbb{Z}}^{d}$, we have
$\sum_{k=1}^{\infty}\,\mu(Z_{k}={\underline{\ell}})<+\infty$.

Therefore, the series
$I({\underline{\ell}}):=-1_{{\underline{\ell}}={\underline{0}}}+\sum_{k=0}^{\infty}\,[\mu(Z_{k}={\underline{\ell}})+\mu(Z_{k}=-{\underline{\ell}})]$
converges and by (43) and Lemma 3.1, the even functions
$\frac{1}{n}K_{n}(x,.)$ satisfy:

$\displaystyle\int_{{\mathbb{T}}^{d}}\frac{1}{n}K_{n}(x,.)\,\cos
2\pi\langle{\underline{\ell}},.\rangle\,d{\underline{t}}$
$\displaystyle=-1_{{\underline{\ell}}={\underline{0}}}+\sum_{k=0}^{n-1}\,\frac{1}{n}\sum_{j=0}^{n-k-1}[1_{Z_{k}(T^{j}x)={\underline{\ell}}}+1_{Z_{k}(T^{j}x)=-{\underline{\ell}}}]{\underset{n\to\infty}{\to}}\
I({\underline{\ell}}).$

Note that above the sum over $k$ is written starting from 0. By letting $n$
tend to infinity in the relation

$\displaystyle-1_{{\underline{\ell}}={\underline{0}}}+\sum_{k=0}^{\infty}\,\lambda^{k}[\mu(Z_{k}={\underline{\ell}})+\mu(Z_{k}=-{\underline{\ell}})]$
$\displaystyle=$ $\displaystyle\int_{{\mathbb{T}}^{d}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,[-1+2\Re e({1\over
1-\lambda\Psi(.)})]\,d{\underline{t}}\ =\int_{{\mathbb{T}}^{d}}\cos
2\pi\langle{\underline{\ell}},{\underline{t}}\rangle\,\Phi_{\lambda}(.)\,d{\underline{t}},$

we get since the left sum tends to $I({\underline{\ell}})$:

$\displaystyle I({\underline{\ell}})=\lim_{\lambda\uparrow
1}\int_{{\mathbb{T}}^{d}}\cos
2\pi\langle{\underline{\ell}},{\underline{t}}\rangle\,\Phi_{\lambda}({\underline{t}})\,d{\underline{t}}.$

Taking ${\underline{\ell}}={\underline{0}}$ in the previous formula, it
follows from Fatou’s lemma:

$\displaystyle
I({\underline{0}})=1+2\sum_{k=1}^{\infty}\,\mu(Z_{k}={\underline{0}})=\lim_{\lambda\uparrow
1}\int_{{\mathbb{T}}^{d}}\,\Phi_{\lambda}({\underline{t}})\,d{\underline{t}}\geq\int_{{\mathbb{T}}^{d}}\lim_{\lambda\uparrow
1}\Phi_{\lambda}({\underline{t}})\,d{\underline{t}}=\int_{{\mathbb{T}}^{d}}\Phi({\underline{t}})\,d{\underline{t}}.$

This shows the integrability of $\Phi$ on ${\mathbb{T}}^{d}$ and we can write
with a constant $M\geq 0$

$\displaystyle I({\underline{0}})=\lim_{\lambda\uparrow
1}\int_{{\mathbb{T}}^{d}}\,\Phi_{\lambda}({\underline{t}})\,d{\underline{t}}=\int_{{\mathbb{T}}^{d}}\lim_{\lambda\uparrow
1}\Phi_{\lambda}({\underline{t}})\,d{\underline{t}}+M=\int_{{\mathbb{T}}^{d}}\Phi({\underline{t}})\,d{\underline{t}}+M.$

Let $U_{\eta}$ be the ball of radius $\eta>0$ centered at ${\underline{0}}$.
By aperiodicity of the r.w., $\Psi({\underline{t}})\not=1$ for
${\underline{t}}$ in $U_{\eta}^{c}$, the complementary in ${\mathbb{T}}^{d}$
of $U_{\eta}$, This implies $\sup_{{\underline{t}}\in
U_{\eta}^{c}}\sup_{\lambda<1}\Phi_{\lambda}({\underline{t}})<+\infty$.

Therefore, we get: $\displaystyle\lim_{\lambda\uparrow
1}\int_{U_{\eta}^{c}}\cos
2\pi\langle{\underline{\ell}},{\underline{t}}\rangle\,\Phi_{\lambda}({\underline{t}})\,d{\underline{t}}=\int_{U_{\eta}^{c}}\cos
2\pi\langle{\underline{\ell}},{\underline{t}}\rangle\,\Phi({\underline{t}})\,d{\underline{t}}$,
hence:

$\displaystyle I({\underline{\ell}})=\int_{U_{\eta}^{c}}\cos
2\pi\langle{\underline{\ell}},{\underline{t}}\rangle\,\Phi({\underline{t}})\,d{\underline{t}}+\lim_{\lambda\uparrow
1}\int_{U_{\eta}}\cos
2\pi\langle{\underline{\ell}},{\underline{t}}\rangle\,\Phi_{\lambda}({\underline{t}})\,d{\underline{t}},\,\forall\eta>0,$

which can be be written:

(45) $\displaystyle-\int_{U_{\eta}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi\,d{\underline{t}}=I({\underline{\ell}})-\int_{{\mathbb{T}}^{d}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi\,d{\underline{t}}-\lim_{\lambda\uparrow
1}\int_{U_{\eta}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi_{\lambda}\,d{\underline{t}}.$

Let $\varepsilon>0$. By positivity of $\Phi_{\lambda}$, we have, for
$\eta(\varepsilon)$ small enough:

$\displaystyle(1-\varepsilon)\int_{U_{\eta(\varepsilon)}}\,\Phi_{\lambda}\,d{\underline{t}}\leq\int_{U_{\eta(\varepsilon)}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi_{\lambda}\,d{\underline{t}}\leq(1+\varepsilon)\int_{U_{\eta(\varepsilon)}}\,\Phi_{\lambda}\,d{\underline{t}};$

By subtracting $\int_{U_{\eta}(\varepsilon)}\cos
2\pi\langle{\underline{\ell}},{\underline{t}}\rangle\,\Phi({\underline{t}})\,d{\underline{t}}$
in the previous inequalities and (45), we get:

$\displaystyle(1-\varepsilon)\int_{U_{\eta(\varepsilon)}}\,\Phi_{\lambda}\,d{\underline{t}}-\int_{U_{\eta(\varepsilon)}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi\,d{\underline{t}}$
$\displaystyle\leq I({\underline{\ell}})-\int_{{\mathbb{T}}^{d}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi\,d{\underline{t}}-\lim_{\lambda\uparrow
1}\int_{U_{\eta}(\varepsilon)}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi_{\lambda}\,d{\underline{t}}+\int_{U_{\eta(\varepsilon)}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi_{\lambda}\,d{\underline{t}}$
$\displaystyle\leq(1+\varepsilon)\int_{U_{\eta(\varepsilon)}}\,\Phi_{\lambda}\,d{\underline{t}}-\int_{U_{\eta(\varepsilon)}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi\,d{\underline{t}};$

As we can chose $\lambda$ such that

$|-\lim_{\lambda\uparrow 1}\int_{U_{\eta}(\varepsilon)}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi_{\lambda}\,d{\underline{t}}+\int_{U_{\eta(\varepsilon)}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi_{\lambda}\,d{\underline{t}}|\leq\varepsilon,$

we obtain:

$\displaystyle-\varepsilon+(1-\varepsilon)\int_{U_{\eta(\varepsilon)}}\,\Phi_{\lambda}\,d{\underline{t}}-\int_{U_{\eta(\varepsilon)}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi\,d{\underline{t}}$
$\displaystyle\leq I({\underline{\ell}})-\int_{{\mathbb{T}}^{d}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi\,d{\underline{t}}\leq\varepsilon+(1+\varepsilon)\int_{U_{\eta(\varepsilon)}}\,\Phi_{\lambda}\,d{\underline{t}}-\int_{U_{\eta(\varepsilon)}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi\,d{\underline{t}}$

For $\varepsilon$ small enough, $\int_{U_{\eta(\varepsilon)}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi\,d{\underline{t}}$ can be made
arbitrary small, as well as
$\varepsilon\sup_{\lambda<1}\int_{U_{\eta}}\Phi_{\lambda}\,d{\underline{t}}$,
since $\Phi$ is integrable and
$\sup_{\lambda<1}\int_{{\mathbb{T}}^{d}}\Phi_{\lambda}\,d{\underline{t}}<\infty$.

This shows that $I({\underline{\ell}})-\int_{{\mathbb{T}}^{d}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi\,d{\underline{t}}-\int_{U_{\eta(\varepsilon)}}\,\Phi_{\lambda}\,d{\underline{t}}$
can be made arbitrarily small for $\varepsilon>0$ small and $\lambda$ close to
1. The same is true for ${\underline{\ell}}=0$ and also for the difference

$\displaystyle[I({\underline{\ell}})-\int_{{\mathbb{T}}^{d}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi\,d{\underline{t}}-\int_{U_{\eta(\varepsilon)}}\,\Phi_{\lambda}\,d{\underline{t}}]-[I({\underline{0}})-\int_{{\mathbb{T}}^{d}}\,\Phi\,d{\underline{t}}-\int_{U_{\eta(\varepsilon)}}\,\Phi_{\lambda}\,d{\underline{t}}]$
$\displaystyle=[I({\underline{\ell}})-\int_{{\mathbb{T}}^{d}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi\,d{\underline{t}}]-[I({\underline{0}})-\int_{{\mathbb{T}}^{d}}\,\Phi\,d{\underline{t}}]=[I({\underline{\ell}})-\int_{{\mathbb{T}}^{d}}\cos
2\pi\langle{\underline{\ell}},.\rangle\,\Phi\,d{\underline{t}}]-M].$

Therefore $I({\underline{\ell}})=\int_{{\mathbb{T}}^{d}}\cos
2\pi\langle{\underline{\ell}},{\underline{t}}\rangle\,\Phi({\underline{t}})\,d{\underline{t}}+M$
for all ${\underline{\ell}}$ and the Fourier coefficients of
$\frac{1}{n}K_{n}$ converges to those of $\Phi+M\delta_{\underline{0}}$.

As the non-negative sequence $(\frac{1}{n}K_{n})$ is bounded in $L^{1}$-norm
and the density $\rho$ is continuous, this proves
$\displaystyle\int\frac{1}{n}K_{n}\rho\,d{\underline{t}}\to\int\Phi\rho\,d{\underline{t}}+M\rho({\underline{0}})$.
Moreover, the limit is $>0$ since both $\Phi$ and $\rho$ are not 0 a.e.

It is shown in [28] that $M=0$ for $d>1$. ∎

Behaviour of $M_{n}(x)$.

In the transient case ($d\geq 3$) (at least for a simple r.w.), Erdös and
Taylor (1960) proved that for a constant $\gamma>0$ depending on the
dimension,

$\lim_{n}{M_{n}(x)\over\log n}=\gamma.$

Recurrent case

In dimension 1, H. Kesten has shown that
$\displaystyle\limsup_{n}{M_{n}\over\sqrt{n\,\ln\ln n}}=\sqrt{2}/\sigma$.
Therefore in dimension 1, we have the following lower and upper bounds for
$V_{n}$:

$C_{1}(x)\,n^{\frac{3}{2}}\,(\ln\ln\,n)^{-\frac{1}{2}}\leq V_{n}(x)\leq
C_{2}(x)\,n^{\frac{3}{2}}(\ln\ln n)^{\frac{1}{2}}.$

Dimension $d=2$.

There is a deterministic rate (law of large numbers): for a constant $C_{0}$.

${\int V_{n}\,d\mu\over n\log n}\to C_{0}\text{ and }{V_{n}(x)\over n\log
n}\to C_{0},\text{ for a.e. }x.$

For a planar simple random walk, Erdös and Taylor [16] have shown:

(46) $\displaystyle\limsup_{n}\,{M_{n}(x)\over(\log n)^{2}}\leq\frac{1}{\pi}.$

The result has been extended by Dembo, Peres, Rosen and Zeitouni [15], who
proved for an aperiodic centered random walk on ${\mathbb{Z}}^{2}$ with
moments of all orders:

$\displaystyle\lim_{n}{M_{n}(x)\over(\log
n)^{2}}=\frac{1}{2\pi\det(\Gamma)^{\frac{1}{2}}},$

where $\Gamma$ is the covariance matrix associated to the random walk.

As shown in the proof in [15], it suffices to suppose that the 2-dimensional
r.w. is aperiodic. Moreover, the proof for the upper bound is based on the
local limit theorem which uses only the existence of the moment of order 2.
Therefore, assuming the existence of the moment of order 2, the upper bound
(46) holds.

It follows in this case: there exist $C(x)$ a.e finite such that:

$\displaystyle{M_{n}^{2}(x)\over V_{n}(x)}\leq C(x){(\log n)^{3}\over n}.$

###  3.2. Extensions of the r.w. case

1) Consequence of the Local Limit Theorem (LLT).

The Local Limit Theorem, when it is satisfied by the cocycle $(T,f)$, gives
some pointwise information on $V_{n}(x)$. For example, if $d=2$, the following
lemma holds:

######  Lemma 3.3.

Suppose that the LLT holds and $d=2$. Then, for every $\varepsilon>0$, there
is an integrable function $C$ (depending on $\varepsilon$), such that:

(47) $\displaystyle N_{n}(x,0)\leq C(x)(\ln n)^{2+\varepsilon},\ V_{n}(x)\leq
C(x)\,n\,(\log n)^{2+\varepsilon}.$

Proof. By the LLT, it holds, for $n\geq 1$,

$\int N_{n}(.,0)d\mu=\sum_{k=1}^{n}\int 1_{f_{k}=0}\,d\mu\leq
C\sum_{k=1}^{n}{1\over k}\leq C\ln n.$

Let $\varepsilon$ be a positive constant. Putting
$\Gamma(x)=\sum_{n=1}^{\infty}n^{-(2+\varepsilon)}N_{2^{n}}(x,0)$, we have:

$\int\Gamma(x)\,d\mu(x)\leq
C\sum_{n=1}^{\infty}n^{-(2+\varepsilon)}n=C\sum_{n=1}^{\infty}n^{-(1+\varepsilon)}<+\infty,$

so that $N_{2^{n}}(x,0)\leq\Gamma(x)\,n^{2+\varepsilon}$, where $\Gamma$ is
integrable.

If $2^{k_{n}}\leq n<2^{(k_{n}+1)}$, then with $p=2+\varepsilon$, we have for
$n$ big enough,

${N_{n}(x,0)\over(\log_{2}n)^{p}}\leq{N_{2^{(k_{n}+1)}}(x,0)\over
k_{n}^{p}}\leq{(k_{n}+1)^{p}\over
k_{n}^{p}}{N_{2^{(k_{n}+1)}}(x,0)\over(k_{n}+1)^{p}}=(1+1/k_{n})^{p}\,\Gamma(x)\leq
2\Gamma(x).$

For $V_{n}(x)$, by (27) we have:

$\int V_{n}(x)\,d\mu(x)=2\sum_{k=1}^{n-1}\int N_{n-k}(x,0)\,d\mu(x)+n=O(n\log
n).$

As above for $N_{n}(x,0)$, the pointwise bound (47) follows. ∎

Among example of cocycles satisfying a LLT, there are the r.w.’s (but with
more precise results as recalled above), but also cocycles generated by
functions with values in ${\mathbb{Z}}^{d}$ depending on a finite number of
coordinates over a sub-shift of finite type endowed with a Gibbs measure
([19]), ([18]).

2) Functions depending on a finite number of coordinates on a Bernoulli
scheme.

Now we try to bound $M_{n}(x)$ in situation which extends slightly that of
random walks.

Suppose that $(X,\mu,T)$ is a Bernoulli scheme with $X=I^{\mathbb{N}}$, where
$I$ is a finite set. Let $f:x\to f(x_{1},...,x_{r})$ be a centered function
from $X$ to ${\mathbb{Z}}^{d}$, $d\geq 1$, depending on a finite number of
coordinates.

Let us consider the generalized random walk $(Z_{n})$ defined by the sequence
of ergodic sums $Z_{n}(x)=f_{n}(x)=X_{0}(x)+...+X_{n-1}(x)$, where
$X_{k}(x)=f(T^{k}x)$.

######  Lemma 3.4.

For all $m\geq 1$ and for constants $C_{m},C_{m}^{\prime}$ independent of
${\underline{\ell}}$, we have:

$\displaystyle\int
N_{n}^{m}(.,{\underline{\ell}})\,d\mu=\int[\sum_{k=1}^{n}1_{f_{k}={\underline{\ell}}}]^{m}\,d\mu$
$\displaystyle\leq C_{m}n^{m/2},{\rm\ for\ }d=1,$ $\displaystyle\leq
C_{m}^{\prime}({\rm Log}n)^{m},{\rm\ for\ }d=2.$

Proof. We bound the sum $\sum_{1\leq k_{1}<k_{2}<...<k_{m}\leq
n}\mu(f_{k_{1}}={\underline{\ell}},f_{k_{2}}={\underline{\ell}},...,f_{k_{m}}={\underline{\ell}})$.

For $r\leq k_{1}<k_{2}<...<k_{m}\leq n$, writing $X_{k}$ instead of $T^{k}f$,
we have:

$\displaystyle\mu(X_{0}+...+X_{k_{1}-1}={\underline{\ell}},X_{k_{1}}+...+X_{k_{2}-1}={\underline{0}},...,X_{k_{m-1}}+...+X_{k_{m}-1}={\underline{0}})$
$\displaystyle=\sum_{a_{1,1},...,a_{r,1},...,a_{1,m},...,a_{r,m}\in S}\ \mu[$
$\displaystyle
X_{0}+...+X_{k_{1}-r}={\underline{\ell}}-(a_{1,1}+...+a_{r,1}),X_{k_{1}-r+1}=a_{1,1,}...,X_{k_{1}-1}=a_{r,1},$
$\displaystyle
X_{k_{1}}+...+X_{k_{2}-r}=-(a_{1,2}+...+a_{r,2}),X_{k_{2}-r+1}=a_{1,2},...,X_{k_{2}-1}=a_{r,2},...$
$\displaystyle
X_{k_{m-1}}+...+X_{k_{m}-r}=-(a_{1,m}+...+a_{r,m}),X_{k_{m}-r+1}=a_{1,m},...,X_{k_{m}-1}=a_{r,m}];$

which is less than:

$\displaystyle\sum_{a_{1,1},...,a_{r,1},...,a_{1,m},...,a_{r,m}\in S}\
\mu[X_{0}+...+X_{k_{1}-r}={\underline{\ell}}-(a_{1,1}+...+a_{r,1}),$
$\displaystyle X_{k_{1}}+...+X_{k_{2}-r}=-(a_{1,2}+...+a_{r,2}),...,\
X_{k_{m-1}}+...+X_{k_{m}-r}=-(a_{1,m}+...+a_{r,m})].$

Now inside $[.]$ the events are independent. By independence and stationarity,
the preceding sum is

$\displaystyle\sum_{a_{1,1},...,a_{r,1},...,a_{1,m},...,a_{r,m}\in S}$
$\displaystyle\mu[X_{0}+...+X_{k_{1}-r}={\underline{\ell}}-(a_{1,1}+...+a_{r,1})]$
$\displaystyle\mu[X_{0}+...+X_{k_{2}-k_{1}-r}=-(a_{1,2}+...+a_{r,2})]\ ...$
$\displaystyle\mu[\ X_{0}+...+X_{k_{m}-k_{m-1}-r}=-(a_{1,m}+...+a_{r,m})].$

With
$\tau_{n}(k)=\sup_{\underline{\ell}}\mu(X_{0}+...+X_{k-1}={\underline{\ell}})\,1_{[0,n](k)}$,
we get the following bound

$\displaystyle\sum_{1\leq k_{1}<k_{2}<...<k_{m}\leq
n}\mu(f_{k_{1}}={\underline{\ell}},f_{k_{2}}={\underline{\ell}},...,f_{k_{m}}={\underline{\ell}})$
$\displaystyle=\sum_{r\leq k_{1}<k_{2}<...<k_{m}\leq
n}\mu(X_{1}+...+X_{k_{1}-1}={\underline{\ell}},X_{k_{1}}+...+X_{k_{2}-1}={\underline{0}},...,X_{k_{m-1}}+...+X_{k_{m}-1}={\underline{0}})$
$\displaystyle\leq s^{rm}\,\sum_{r\leq k_{1}<k_{2}<...<k_{m}\leq
n}\tau_{n}({k_{1}-r})\,\tau_{n}({k_{2}-k_{1}-r})\,...\,\tau_{n}({k_{m}-k_{m-1}-r})$
$\displaystyle\leq Cs^{rm}\sum_{k}(\tau_{n}*\tau_{n}*...*\tau_{n})(k)\leq
Cs^{rm}(\sum_{k}\tau_{n}(k))^{m}.$

We have $s^{rm}$ terms for the first sum, where the cardinal $|S|$ is denoted
by $s$,.

Now we can use, as for the usual r.w., convolution and the local limit
theorem. ∎

From the lemma, it follows easily in the recurrent case that for a.e. $x$, for
all $\varepsilon>0$,

$\text{ if }d=1,M_{n}(x)=o(n^{\frac{1}{2}+\varepsilon})\ \text{and, if
}d=2,M_{n}(x)=o(n^{\varepsilon}).$

In the transient case, if there is a moment of order $\eta$ for some $\eta>0$,
then $M_{n}(x)=o(n^{\varepsilon})$ for all $\varepsilon>0$. For these
estimates, in both cases, see [11].

A question if to extend the previous results to a larger class of functions
depending weakly on the far coordinates. For such an extension is that
explicit bounds in the LLT are not always available.

###  3.3. Step functions over rotations

Now we take $X=\mathbb{T}^{r}$, $r\geq 1$ endowed with $\mu$, the uniform
measure and we consider cocycles over rotations. When they are centered, such
cocycles are strongly recurrent and therefore the associated quantities
$V_{n}$ and $M_{n}$ are big. The difficult part is to bound them from above.
We will give an example where an upper bound can be obtained.

Let $T_{\alpha}$ be the rotation by an irrational $\alpha$. For
$f:X\to{\mathbb{Z}}^{d}$, recall that the cylinder map (cf. Subsection 2.1) is
$\tilde{T}_{f,\alpha}=\tilde{T}_{\alpha}:X\times{\mathbb{Z}}^{d}\to
X\times{\mathbb{Z}}^{d}$ defined by
$\tilde{T}_{\alpha}(x,{\underline{\ell}})=(x+\alpha,{\underline{\ell}}+f(x))$.

Non centered step cocycles over a rotation.

Let $f$ be a non centered function with a finite number of values values in
${\mathbb{Z}}^{d}$. Suppose that $f$ is Riemann integrable, which amounts to
assume that, for the uniform measure of the torus, the measure of the set of
discontinuity points of $f$ is zero.

Then by a remark in Subsection 2.2, $M_{n}(x)$ is bounded uniformly in $x$ and
$n$. Therefore, for $V_{n}(x)$, the bounds $n\leq V_{n}(x)\leq Cn$ are
satisfied.

Centered step cocycles over a 1-dimensional rotation.

The interesting situation is that of centered functions. We will consider the
case $r=1$ and when the irrational number $\alpha$ has bounded partial
quotients.

Recall that an irrational $\alpha$ with continued fraction expansion
$[0;a_{1},a_{2},...,a_{n},...]$ is said to have bounded partial quotients
(bpq) if $\sup_{n}a_{n}<+\infty$. The set of bpq numbers has Lebesgue measure
zero and Hausdorff dimension 1.

In the sequel of this subsection, $\alpha$ will be an irrational bpq number
(for instance a quadratic irrational) and $f$ a centered function with values
in ${\mathbb{Z}}$ and bounded variation.

By Denjoy-Koksma inequality, there is a logarithmic bound for the cocycle
$(T_{\alpha},f)$: $|f_{n}(x)|\leq C\ln n$, for a constant $C$.

The cocycle is strongly recurrent to 0 (and this is true for $d\geq 1$ if $f$
centered has values in ${\mathbb{Z}}^{d}$, when its components have bounded
variation). This makes the corresponding maximum $M_{n}(x)$ big. Nevertheless,
we will see that condition (18) is satisfied, at least for a special example.

Lower bound.

Lower bound for $V_{n}$ and variance, case $d=1$.

For a general sequence $(z_{k})$, we can obtain a lower bound for $V_{n}$ by
an elementary method when there is an upper bound for the variance defined
below.

######  Lemma 3.5.

Defining the mean $m_{n}$ and the variance $\sigma_{n}^{2}$ by

$m_{n}=\frac{1}{n}\sum_{k=1}^{n}z_{k},\
\sigma_{n}^{2}=\frac{1}{n}\sum_{k=1}^{n}(z_{k}-m_{n})^{2},$

we have

(48) $\displaystyle V_{n}\geq\frac{1}{9}\,{n^{2}\over\sigma_{n}},\text{ if
}\sigma_{n}>1.$

Proof. Suppose that $\sigma_{n}>0$. For $\lambda>1$, let
$\Delta_{\lambda}:=[-\lambda\sigma_{n}+m_{n},\,\lambda\sigma_{n}+m_{n}]\bigcap{\mathbb{Z}}$.
We have:

$\sigma_{n}^{2}\geq\frac{1}{n}\sum_{k=0}^{n-1}(z_{k}-m_{n})^{2}1_{z_{k}\in\Delta_{\lambda}^{c}}\geq\frac{1}{n}\sum_{k=0}^{n-1}(1_{z_{k}\in\Delta_{\lambda}^{c}})\,\lambda^{2}\sigma_{n}^{2}.$

Therefore: $\sum_{k=0}^{n-1}1_{z_{k}\in\Delta_{\lambda}}\geq
n(1-\lambda^{-2})$. As ${\rm Card}(\Delta_{\lambda})\leq
2\lambda\sigma_{n}+1$. It follows by (24):

$V_{n}\geq{(1-\lambda^{-2})^{2}\over 2\lambda\sigma_{n}+1}\,n^{2}.$

For $\lambda=2$ we get: $V_{n}\geq\ {\frac{9}{16}\over
4\sigma_{n}+1}\,n^{2}\geq\frac{9}{80}\,{n^{2}\over\sigma_{n}},\text{ if
}\sigma_{n}>1$; hence (48). ∎

If $z_{k}$ is given by ergodic sums, i.e., $z_{k}=f_{k}(x)$ , let

$m_{n}(x):=\frac{1}{n}\sum_{k=1}^{n}f_{k}(x),\
\sigma_{n}^{2}(x)=\frac{1}{n}\sum_{k=1}^{n}(f_{k}(x)-m_{n}(x))^{2}.$

By [5, Proposition 13], for $\alpha$ bpq and $f$ with bounded vartion, it
holds $\sigma_{n}^{2}(x)\leq C\ln n$.

Using (48) and $V_{n}(x)\leq nM_{n}(x)$, this gives a lower bound for
$V_{n}(x)$ and $M_{n}(x)$:

(49) $\displaystyle V_{n}(x)\geq c\,{n^{2}\over\sqrt{\ln n}},\ M_{n}(x)\geq
c\,{n\over\sqrt{\ln n}}.$

Below we will get an estimate from above in the following example.

######  Example 3.6.

$f={\bf 1}_{[0,\frac{1}{2})}-{\bf 1}_{[\frac{1}{2},1)}$ and $\alpha$ bpq.

Upper bound for the example (3.6).

For $f$ as above and $\alpha$ bpq, we have by [1], for some constant
$C_{1}>0$,

(50) $\displaystyle\|N_{n}(\cdot,0)\|_{\infty}=\|\tilde{S}_{n}({\bf
1}_{\mathbb{T}^{1}\times\\{0\\}})(\cdot,0)\|_{\infty}\leq\frac{C_{1}n}{\sqrt{\log
n}}.$

Remark that the bound (50) is obtained in [1] as the limit of
$\|N_{n}(\cdot,0)\|_{p}$, the $L^{p}$-norm of $N_{n}(\cdot,0)$, as $p$ goes to
$\infty$. Therefore the bound holds for the norm $\|.\|_{ess\,sup}$, but it
can be easily replaced by the uniform norm as written above. Indeed, for any
$x$, there is a neighborhood $V(x)$ of $x$, such that for $y\in V(x)$,
$|N_{n}(x,0)-N_{n}(y,0)|\leq 1$ (at most one jump in $V(x)$). As one can find
$y\in V(x)$ satisfying $N_{n}(y,0)\leq\frac{C_{1}n}{\sqrt{\log n}}$, the same
inequality holds for $x$, with $C_{1}$ replaced by $2C_{1}$.

Using Remark 2.4, it follows:

(51) $\displaystyle M_{n}(x)\leq C_{1}\frac{n}{\sqrt{\log n}}.$

By (51) and since $V_{n}(x)\leq n\,M_{n}(x)$, we obtain

(52) $\displaystyle V_{n}(x)\leq C_{1}\frac{n^{2}}{\sqrt{\log n}}.$

From (49), (51) and (52), it follows: $V_{n}(x)\asymp n^{2}/\sqrt{\log n}$ and
$M_{n}(x)\asymp n/\sqrt{\log n}$, where $a_{n}\asymp b_{n}$ for two sequences
$(a_{n})$ and $(b_{n})$ means $c\,a_{n}\leq b_{n}\leq C\,a_{n},\forall n\geq
1$, with two positive constants $c,C$.

Therefore we get in this special example 3.6:

(53) $\displaystyle{M_{n}^{2}(x)\over V_{n}(x)}\leq(\frac{C_{1}n}{\sqrt{\log
n}})^{2}/\frac{cn^{2}}{\sqrt{\log n}}={C_{1}^{2}\over c}\,\frac{1}{\sqrt{\log
n}}\to 0.$

Condition (18) of Theorem 1.9 is satisfied in this example, as well as the
condition of Theorem 1.6 a), hence a Glivenko-Cantelli theorem along
$(S_{n}f(x))$ for i.i.d. r.v.’s.

But the sufficient conditions for the Glivenko-Cantelli theorems 1.5, 1.6 b),
1.8 are not satisfied by this cocycle and more generally, in view of the lower
bound (49), by a cocycle defined by step functions over a bpq irrational
rotation.

##  4\. About limit theorems along ergodic sums

###  4.1. Glivenko-Cantelli theorem along ergodic sums

The Glivenko-Cantelli theorem recalled in the introduction is a (pointwise)
law of large numbers uniform over a set of functions (here the indicators of
intervals). When the r.v.’s $X_{k}$ are i.i.d., the proof is an easy
consequence of the strong law of large numbers applied to the sequence of
i.i.d. bounded r.v.’s ($1_{X_{k}\leq s})$. Using Birkhoff’s ergodic theorem,
the Glivenko-Cantelli theorem has been extended to the setting of a strictly
stationary sequence $(X_{k})$ of random variables. More precisely, formulated
in terms of dynamical systems, the following holds:

Let $(Y,{\cal A},\nu)$ be a probability space and $S$ an ergodic measure
preserving transformation on $Y$. For any measurable function
$\varphi:Y\to{\mathbb{R}}$, let us consider the strictly stationary sequence
$(X_{k})$ defined by $X_{k}=\varphi\circ S^{k},k\geq 0$. Then the sequence of
empirical distribution functions satisfies: $\text{ for }\nu\text{ a.e. }y\in
Y,\sup_{s}|\frac{1}{n}\sum_{k=0}^{n-1}1_{X_{(}y)k\leq s}-F(s)|\to 0$, where
$F(s)=\nu(\varphi\leq s)$.

Observe that the result is an application of Birkhoff’s theorem and Lemma 1.1
recalled in Section 1. Its extension to the non ergodic case has been
formulated by Tucker [29], the distribution function $F(s)$ being replaced by
the conditional distribution function ${\mathbb{E}}(1_{\varphi\leq
s}|{\mathcal{J}})$, where ${\cal J}$ is the $\sigma$-algebra of $S$-invariant
sets. In others words, we have:

$\text{ for }\nu\text{ a.e. }y\in Y,\
\lim_{n\to\infty}\,\sup_{s}|\frac{1}{n}\sum_{k=0}^{n-1}1_{\varphi(S^{k}y)\leq
s}-{\mathbb{E}}(1_{\varphi\leq s}|{\mathcal{J}})(y)|=0.$

The above formula relies on the ergodic decomposition which can be used in the
proof.

In the previous framework, for a process, a Glivenko-Cantelli like theorem
sampled along a sequence generated by a dynamical system can be obtained as
follows:

As in Subsection 2, let $T$ be an ergodic measure preserving transformation on
a probability space $(X,{\cal B},\mu)$ and $f$ a measurable function on $X$
with values in ${\mathbb{Z}}^{d}$, $d\geq 1$.

Let us take a second system $(\Omega,{\mathbb{P}},\theta)$, where
$\theta=(\theta^{\underline{\ell}})_{{\underline{\ell}}\in{\mathbb{Z}}^{d}}$
is a ${\mathbb{Z}}^{d}$-action preserving ${\mathbb{P}}$.

The skew product associated to the cocycle $(T,f)$ and $\theta$ is the map:
$T_{\theta,f}:(x,\omega)\to(Tx,\theta^{f(x)}\omega)$ from $X\times\Omega$ to
itself. By iteration we get:

$T_{\theta,f}^{k}(x,\omega)=(T^{k}x,\theta^{f_{k}(x)}\omega).$

For example, as ${\mathbb{Z}}^{d}$-action, we can take a
${\mathbb{Z}}^{d}$-Bernoulli shift
$(\Omega,{\mathbb{P}},(\theta^{\underline{\ell}})_{{\underline{\ell}}\in{\mathbb{Z}}^{d}})$,
with ${\mathbb{P}}$ a product measure and $\theta$ the shift on the
coordinates. If $X_{0}$ is the first coordinate map, then
$(X_{\underline{\ell}})=(X_{0}\circ\theta^{\underline{\ell}})$ is a family of
i.i.d. r.v.’s indexed by ${\mathbb{Z}}^{d}$.

In general, let ${\mathcal{I}}_{\theta,f}$ denote the conditional expectation
with respect to the $\sigma$-algebra of $T_{\theta,f}$-invariant sets. The
ergodic theorem for $T_{\theta,f}$ shows that, for $\psi\in
L^{1}(\mu\times{\mathbb{P}})$,

(54)
$\displaystyle\lim_{n}\frac{1}{n}\,\sum_{k=0}^{n-1}\psi(T^{k}x,\theta^{f_{k}(x)}\omega)={\mathcal{I}}_{\theta,f}(\psi)(x,\omega),\text{
for }\mu\times{\mathbb{P}}\text{-a.e.}(x,\omega).$

If $\varphi$ is a measurable function on $\Omega$, putting
$\psi_{s}(x,\omega)={\bf 1}_{I_{s}}(\varphi(\omega))$, where $I_{s}$ is the
half-line $]-\infty,s]$, we have

$\psi_{s}(T_{\theta,f}^{k}(x,\omega))={\bf
1}_{I_{s}}(\varphi(\theta^{f_{k}(x)}\omega)).$

By the quoted Tucker’s result, the convergence in (54) for each $\psi_{s}$,
$s\in{\mathbb{R}}$, can be strengthened into a uniform convergence with
respect to $s$:

$\displaystyle\text{ for $\mu\times{\mathbb{P}}$-a.e
}(x,\omega),\,\frac{1}{n}\,\sup_{s}|\sum_{k=0}^{n-1}\,{\bf
1}_{I_{s}}(\varphi(\theta^{f_{k}(x)}\omega))-\mathcal{I}(\psi_{s})(x,\omega)|\to
0.$

Therefore, by the Fubini theorem, there is a “sampled” version of the
Glivenko-Cantelli theorem for the empirical process of a stationary sequence:

######  Proposition 4.1.

For $\mu$-a.e $x$, we have

$\displaystyle|\sup_{s}\frac{1}{n}\,\sum_{k=0}^{n-1}\,{\bf
1}_{I_{s}}(\varphi(\theta^{f_{k}(x)}\omega))-\mathcal{I}(\psi_{s})(x,\omega)|\to
0,\text{ for ${\mathbb{P}}$-a.e }\omega.$

When $T_{\theta,f}$ is ergodic, if $\psi\in L^{1}(\mu\times{\mathbb{P}})$, we
have ${\mathcal{I}}_{\theta,f}(\psi)(x,\omega)=\int\psi\,d\mu\,d{\mathbb{P}}$,
for $\mu\times{\mathbb{P}}\text{-a.e.}\,(x,\omega)$, and the centering
$\mathcal{I}(\psi_{s})(x,\omega)$ is given by the distribution function
$F(s)=\mu(\varphi\leq s)$. In this case, for a.e. $x$, a Glivenko-Cantelli
theorem with the usual centering holds for the empirical process sampled along
the sequence $(z_{n})$ given by $z_{n}=S_{n}f(x)$ (with a set of $\omega$’s of
${\mathbb{P}}$-measure 1 depending on $x$).

The lemma below shows, as it is known, that ergodicity of the cylinder map
$\tilde{T}_{f}$ implies ergodicity of the skew map $T_{\theta,f}$. Let us
sketch a proof.

######  Lemma 4.2.

Suppose that the cocycle $(T,f)$ is recurrent and the map $\tilde{T}_{f}$
ergodic. If the action of ${\mathbb{Z}}^{d}$ by $\theta$ on
$(\Omega,{\mathbb{P}})$ is ergodic, then $T_{\theta,f}$ is ergodic on
$(X\times\Omega,\mu\times{\mathbb{P}})$.

Proof. : Let $\Phi$ be a $T_{\theta,f}$ invariant measurable function on
$X\times\Omega$:

$\Phi(Tx,\theta^{f(x)}\omega)=\Phi(x,\omega),\text{ for a.e. }(x,\omega).$

For a.e. $x$, there is a set $\Omega_{x}^{0}$ of full ${\mathbb{P}}$-measure
in $\Omega$ such that $\Phi(Tx,\theta^{f(x)}\omega)=\Phi(x,\omega)$, for all
$\omega\in\Omega_{x}^{0}$. As ${\mathbb{Z}}^{d}$ is countable, for a.e. $x$,
there is a set $\Omega_{x}$ of full measure such that

$\Phi(Tx,\theta^{f(x)}\theta^{\underline{\ell}}\omega)=\Phi(x,\theta^{\underline{\ell}}\omega),\text{
for all }\omega\in\Omega_{x}.$

Let $\omega\in\Omega_{x}$. The function
$\varphi_{\omega}(x,{\underline{\ell}}):=\Phi(x,\theta^{\underline{\ell}}\omega)$
on $X\times{\mathbb{Z}}^{d}$ is measurable, $\tilde{T}_{f}$-invariant:

$\displaystyle\varphi_{\omega}(\tilde{T}_{f}(x,{\underline{\ell}}))$
$\displaystyle=$
$\displaystyle\varphi_{\omega}(Tx,{\underline{\ell}}+f(x))=\Phi(Tx,\theta^{{\underline{\ell}}+f(x)}\omega)$
$\displaystyle=$
$\displaystyle\Phi(Tx,\theta^{f(x)}\theta^{\underline{\ell}}\omega)=\Phi(x,\theta^{\underline{\ell}}\omega)=\varphi_{\omega}(x,{\underline{\ell}}).$

It follows from the ergodicity of $\tilde{T}_{f}$ that there is a constant
$c_{\omega}$ such that $\varphi_{\omega}(x,{\underline{\ell}})=c_{\omega}$ for
a.e. $x$. Therefore $\Phi$ coincides a.e. with a function $\psi$ on $\Omega$
which is $\theta$-invariant, hence a constant by the assumption of ergodicity
of the action of ${\mathbb{Z}}^{d}$ on $\Omega$. ∎

With Fubini’s argument, we get a Glivenko-Cantelli theorem for a.e. $x$, if we
can show that the skew map $T_{\theta,f}$ is ergodic.

There are many examples cylinder flows $\tilde{T}_{f}$ which are shown to be
ergodic in the literature and so providing examples via Lemma 4.2. For
instance, we can take for $T$ an irrational rotation and $f={\bf
1}_{[0,\frac{1}{2})}-{\bf 1}_{[\frac{1}{2},1)}$. The cocycle ($T,f)$ is
ergodic and the above version of Glivenko-Cantelli theorem applies for any
stationary sequence $(X_{k})$ (with a conditional distribution if the
stationary sequence is not ergodic). See also examples for which the skew map
is ergodic in [25].

###  4.2. Discussion: universal sequences

The weakness in the approach of the previous subsection for a sampled
Glivenko-Cantelli theorem along ergodic sums $(S_{k}f(x),k\geq 0)$ is that it
yields a set of $x$’s of $\mu$-measure 1 depending on the dynamical system
$(\Omega,{\mathbb{P}},\theta)$ and on $\varphi$. One can try to reinforce the
statement by introducing a notion of “universal property”.

In this direction, the LLN for sums sampled along ergodic sums is closely
related in the following way to the random ergodic theorems which have been
studied in several papers.

First, let us call “universally good” a sequence $(z_{k})$ such that, for
every dynamical system $(\Omega,{\mathbb{P}},\theta)$, for every $\varphi\in
L^{1}({\mathbb{P}})$, the sequence
$\frac{1}{n}\sum_{k=0}^{n-1}\varphi\circ\theta^{z_{k}}$ converges
${\mathbb{P}}$-a.e.

We say that $(T,f)$ a “(pointwise) good averaging cocycle” (or a universally
representative sampling scheme) if, for $\mu\text{-a.e. }x$, the sequence
$(S_{k}f(x))$ is universally good, i.e., for every dynamical system
$(\Omega,{\mathbb{P}},\theta)$, for every $\varphi\in L^{1}({\mathbb{P}})$,
$\frac{1}{n}\sum_{k=0}^{n-1}\varphi\circ\theta^{S_{k}f(x)}$ converges
${\mathbb{P}}$-a.e.

The definition of a “mean good averaging cocycle” is similar, changing the
above convergence into convergence in $L^{2}({\mathbb{P}})$-norm, for every
$\varphi$ in $L^{2}({\mathbb{P}})$.

A question which has been studied is to find mean or pointwise good averaging
cocycles. In the first direction, examples and counterexamples of mean good
averaging 1-dimensional cocycles are studied in [25],

For pointwise convergence, there are 1-dimensional examples given by cocycles
with a drift. In [23], the following result is shown: the cocycle defined by a
random walk with a moment of order 2 is a pointwise good averaging cocycle if
and only if it is not centered. Moreover it is shown that any ergodic
integrable integer-valued stochastic process with nonzero mean is universally
representative for bounded stationary processes. The proofs are based on the
recurrence time theorem ([6]).

Notice that a related, but different, notion can be introduced by restricting
the dynamical system $(\Omega,{\mathbb{P}},\theta)$ to belong to a given class
${\mathcal{C}}$ of dynamical systems.

Let us call “pointwise good for a class $\cal C$ of dynamical systems”, a
sequence $(z_{k})$ such that, for every dynamical system
$(\Omega,{\mathbb{P}},\theta)$ in the class $\cal C$, for every $\varphi\in
L^{1}({\mathbb{P}})$,
$\lim_{n}\frac{1}{n}\sum_{k=0}^{n-1}\varphi\circ\theta^{z_{k}}=\int\varphi\,d{\mathbb{P}}$,
${\mathbb{P}}$-a.e. There is a similar property for the mean convergence.

This can be also expressed for a class of random fields satisfying a condition
on the decay of correlations.

For example, by Remark 2.8, every cocycle with values in ${\mathbb{Z}}^{d}$
which is not a coboundary is a mean good averaging cocycle for the stationary
r.f.s on ${\mathbb{Z}}^{d}$ such that $\sum_{\underline{\ell}}|\langle
U_{\underline{\ell}},U_{{\underline{0}}}\rangle|<+\infty$.

If $(z_{k})$ is pointwise universally good for a class $\cal C$, clearly we
get the Glivenko-Cantelli property for any dynamical system
$(\Omega,{\mathbb{P}},\theta)$ in $\cal C$ and every measurable function
$\varphi$, i.e.:

(55) $\displaystyle\sup_{s}|\frac{1}{n}\,\sum_{k=0}^{n-1}\,{\bf
1}_{I_{s}}(\varphi(\theta^{z_{k}}\omega))-{\mathbb{P}}(\varphi\leq s)|\to
0,\text{ for ${\mathbb{P}}$-a.e }\omega.$

As we see, there are two different approaches of the notion of universal
sequences for a law of large numbers: either we ask for a LLN along such a
sequence for every dynamical system $(\Omega,{\mathbb{P}},\theta)$ and all
functions in $L^{1}({\mathbb{P}})$ or we fix a class of dynamical systems, or
a class of functions in $L^{1}({\mathbb{P}})$. In the latter case, the
condition on the sequence $(z_{k})$ may be expressed in a quantitative way.
Let us give a known example and recall the proof.

######  Proposition 4.3.

Let $(z_{k})$ be a strictly increasing sequence of positive integers. If the
sequence satisfies: for a finite constant $C$, $z_{k}\leq Ck,\forall k\geq 1$,
then $(z_{k})$ is a pointwise good averaging sequence for the class
$\mathcal{C}$ of dynamical systems $(\Omega,{\mathbb{P}},\theta)$ with
Lebesgue spectrum.

Proof. There is a dense set of functions $\varphi\in L^{1}({\mathbb{P}})$ such
that

(56)
$\displaystyle\frac{1}{n}\sum_{k=0}^{n-1}\varphi(\theta^{z_{k}}\omega)\text{
converges }{\mathbb{P}}\text{-a.e.}$

Indeed, by the SLLN for orthogonal random variables, (56) is satisfied by
$\varphi\in L^{2}({\mathbb{P}})$ such that
$\langle\varphi,\varphi\circ\theta^{k}\rangle=0,\forall k$. The Lebesgue
spectrum property implies that such functions span a dense linear space in
$L^{2}({\mathbb{P}})$, hence in $L^{1}({\mathbb{P}})$.

Moreover, the space of functions $\varphi$ such that (56) holds is closed by
the ergodic maximal lemma in view of the assumption on $(z_{k})$. Therefore
(56) is satisfied by every $\varphi\in L^{1}({\mathbb{P}})$. ∎

To finish, we recall the following example which shows that the behaviour may
depend on the properties of the dynamical system
$(\Omega,{\mathbb{P}},\theta)$ (cf. [13]):

Let $(\Omega,\cal F,{\mathbb{P}})$ be the interval $[0,1]$ endowed with the
Borel $\sigma$-algebra and the Lebesgue measure and take
$f=1_{[0,\frac{1}{2}]}$. Denote by $\cal T$ the class of invertible measure
preserving transformations on this space. It can be shown that there are
increasing sequences $(z_{k})$ of positive integers satisfying the conditions
of the previous proposition such that, for a dense $G_{\delta}$ of elements in
$\cal T$ with continuous spectrum, the ergodic means of $f$ along $(z_{k})$ do
not converge ${\mathbb{P}}$-a.e.

## References

  * [1]
  * [1] Aaronson, J., Bromberg, M. and Nakada, H.: Discrepancy skew products and affine random walks, Israel J. Math. 221 (2017), no. 2, 973-1010.
  * [2] Ambrose L.: Functional generalizations of Hoeffding’s covariance lemma and a formula for Kendall’s tau, Statistics and Probability Letters, 122 (2017), 218-226.
  * [3] Billingsley P.: _Convergence of probability measures_ , 1rst ed. John Wiley & Sons, Inc., 1968.
  * [4] Birkel, T.: A note on the strong law of large numbers for positively dependent random variables, Statist. Probab. Lett. 7 (1988), no. 1, 17-20.
  * [5] Borda, B.: On the distribution of Sudler products and Birkhoff sums for the irrational rotation, Arkiv 2021.
  * [6] Bourgain J., Furstenberg H., Katznelson Y. and Ornstein D.: Appendix on return-time sequences. Publ. Math. IHES, (69) 42-45, 1989.
  * [7] Chung, K.L: _A course in probability theory_ , 3rd ed. (2001), Acad. Press, Inc., San Diego, CA.
  * [8] Chung, K. L., Fuchs, W. H. J. _On the distribution of values of sums of random variables_ , Mem. Amer. Math. Soc. 6 (1951), 157-168.
  * [9] Chung, K. L., Ornstein, D., _On the recurrence of sums of random variables_ , Bull. Amer. Math. Soc. 68 (1962), 30-32.
  * [10] Cohen, G., Conze, J.-P.: On the quenched functional CLT in random sceneries, to appear in Studia Matematica.
  * [11] Cohen, G., Conze, J.-P.: CLT for random walks of commuting endomorphisms on compact abelian groups, J. Theoret. Probab. 30 (2017), no. 1, 143-195.
  * [12] Conze, J. P., Remarques sur les transformations cylindriques et les équations fonctionnelles, Pub. sém. math. info. de Rennes, 1976, fasc. 2, http://www.numdam.org/actas/PSMIR/
  * [13] Conze, J.-P., Convergence des moyennes ergodiques pour des sous-suites, Bull. Soc. Math. Fr., Mémoire 35, (1973), p. 7-15.
  * [14] Deligiannidis G., Gouezel, S., Kosloff, Z.: Boundary of the range of a random walk and the Folner property, Electron. J. Probab. 26: 1-39 (2021). DOI: 10.1214/21-EJP667
  * [15] Dembo, A.; Peres, Y.; Rosen, J.; Zeitouni, O.: Thick points for planar Brownian motion and the Erdos-Taylor conjecture on random walk. Acta Math. 186 (2001), no. 2, 239-270.
  * [16] Erdos, P.; Taylor, S. J.: Some problems concerning the structure of random walk paths. Acta Math. Acad. Sci. Hungar. 11 (1960), 137–162.
  * [17] Esary, J. D.; Proschan, F.; Walkup, D. W., Association of random variables, with applications, Ann. Math. Statist. 38 (1967), 1466-1474.
  * [18] Gouëzel S.: Berry-Esseen theorem and local limit theorem for non uniformly expanding maps, Ann. Inst. H. H. Poincaré PR 41 (2005), 997–1024
  * [19] Guivarc’h, Y. et Hardy, J.: Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov, Ann. Inst. H. Poincaré 24 (1) (1988), 73-98.
  * [20] Hoeffding W.: Probability Inequalities for Sums of Bounded Random Variables, Journal of the AMS vol. 58 (1963), no. 301, 13-30.
  * [21] Hewitt, E.; Ross, K. A.: _Abstract harmonic analysis_ , Vol. II. Die Grundlehren der mathematischen Wissenschaften, Band 152 Springer-Verlag, New York-Berlin 1970.
  * [22] Kesten, H.: An iterated logarithm law for local time. Duke Math. J. 32 (1965), 447-456.
  * [23] Lacey, M.; Petersen, K.; Wierdl, M.; Rudolph, D.: Random ergodic theorems with universally representative sequences., Ann. I. H. Poincaré Prob. Stat. 30 (1994), no. 3, 353-395.
  * [24] Lehmann, E. L.: Some concepts of dependence, Ann. Math. Statist. 37 (1963), 1137-1153.