RANDOMIZED LIMIT THEOREMS

FOR STATIONARY ERGODIC RANDOM PROCESSES AND FIELDS

Youri Davydov

Faculty of Mathematics and Computer Sciences of St. Petersburg State
University, Russia, and Department of Mathematics of Lille University, France.

Arkady Tempelman

Department of Mathematics and Department of Statistics, the Pennsylvania State
University, USA.

###### Abstract

We consider ”randomized” statistics constructed by using a finite number of
observations a random field at randomly chosen points. We generalize the
invariance principle (the functional CLT), the Glivenko–Cantelli theorem, the
theorem about convergence to the Brownian bridge and the Kolmogorov theorem
about the limit distribution of the empirical distribution function, as well
as an improved version of the CLT in A. Tempelman, Randomized multivariate
central limit theorems for ergodic homogeneous random fields, Stochastic
Processes and their Applications. 143 (2022), 89-105. The randomized approach,
introduced in the mentioned work, allows to extend these theorems to all
ergodic homogeneous random fields on $\mathbb{Z}^{m}$ and $\mathbb{R}^{m}.$

_MSC classification_ : Primary 60F05; 60G10; secondary 37A30

_Keywords:_ Central Limit Theorem; Pointwise Ergodic Theorem; stationary
random process; homogeneous random fields; invariance principle; Glivenko-
Cantelli theorem; Brownian bridge, empirical processes.

##  1 Preliminaries

###  1.1 Short review of the article

The article is devoted to the extension of the main Probability limit theorems
to ergodic stationary random processes and to ergodic homogeneous random
fields. Our approach is based on consideration of ”randomized” statistics,
i.e. statistics constructed by using a finite number of observations of a
random field at randomly chosen points, introduced in[62].

In subsection 1.4 and §2 we define finite randomizing sets in the ”time” space
and randomized statistics, calculated on restrictions of the fields to these
sets.

In §3 the fulfillment of a ”randomized” form of the Lindeberg condition is
proved for ergodic homogeneous random fields, possessing the second moment; it
is essentially used in the sequel.

In §4 we consider two kinds of the CLT. Subsection 4.3 is devoted to
”randomized” versions of the classical CLTs which are valid for all ergodic
homogeneous fields; in these theorems the condition $E[|X(0)|^{2}]<\infty$
replaces the condition: $E[|X(0)|^{2+\delta}]<\infty$ for some $\delta$,
imposed by the second author in [62]. The ”randomized” functional central
limit theorem (invariance principle) is considered in subsection 4.5.

In §5 two theorems related to the limiting behavior of the randomized
empirical distribution functions (EDF) are considered: a randomized version of
the Glivenko–Cantelli theorem for multivariate distributions of ergodic
homogeneous random fields and a general theorem devoted the limit of the
distributions of the EDF; as a corollary, the randomized version of the
Kolmogorov convergence theorem for empirical distributions is derived (of
course, these theorems are also valid for ergodic homogeneous random fields).

###  1.2 The ”time” set $T$

In this paper we study random fields defined on a set $T$, which is the
$m$-dimensional Euclidean space $\mathbb{R}^{m}$ or the $m$-dimensional
integer lattice $\mathbb{Z}^{m},\;m\geq 1$ (when $m=1$, the random fields turn
into random processes or random sequences). We denote by $\mathcal{B}$ the
Borel $\sigma$-field on $T$ (if $T=\mathbb{Z}^{m}$, then $\mathcal{B}$
coincides with the collection of all subsets of $\mathbb{Z}^{m}$, and each
function $f$ on $\mathbb{Z}^{m}$ is $\mathcal{B}$-measurable); $\lambda$ is
the Lebesgue measure on $\mathbb{R}^{m}$ and the counting measure on
$\mathbb{Z}^{m}$ (in the latter case $\lambda(A)$ is the cardinality of
$A\subset\mathbb{Z}^{m}$, and $\int_{A}f(t)\lambda(dt)=\sum_{t\in A}f(t)$ if
$\lambda(A)<\infty$).

###  1.3 Random fields

We consider a $d$-dimensional random field
$X(t)=(X^{1}(t),...,X^{d}(t)),\,t\in T,$ over a probability space
$(\Omega_{X},\mathcal{F}_{X},P_{X})$.

Let us recall several definitions. The field $X$ is (strict-
sense)_homogeneous_ if all finite dimensional distributions of $X$ are shift-
invariant:

$P_{X}\\{\omega:X(t_{1}+t,\omega)\in A_{1},...,X(t_{k}+t,\omega)\in A_{k}\\}=$
$P_{X}\\{\omega:X(t_{1},\omega)\in A_{1},...,X(t_{k},\omega)\in A_{k}\\}$

for all $t,t_{i}\in T,\;\;i,k\in\mathbb{N}$ and for all sets $A_{i}$,
belonging to the Borel $\sigma$-field $\mathcal{B}(\mathbb{R}^{d})$.

A family of invertible transformations $\gamma=\\{\gamma_{t},t\in T\\}$ of
$\Omega_{X}$ is said be a _group_ if
$\gamma_{0}\omega\equiv\omega,\gamma_{s+t}=\gamma_{s}\gamma_{t},\gamma^{-1}_{t}=\gamma_{-t}$,
where $s,t\in T,\,\omega\in\Omega_{X}$. The field $X$ _is generated by a
group_ $\gamma$, if $X(t,\omega)=X(0,\gamma_{t}\omega),\,t\in
T,\,\omega\in\Omega_{X}$; since
$X(0,\gamma_{s+t}\omega)=X(0,\gamma_{s}\gamma_{t}\omega)=X(s,\gamma_{t}\omega)$,
this implies:

$X(s+t,\omega)=X(s,\gamma_{t}\omega),s,t\in T.$ (1)

A family $\gamma$ is _measure preserving_ , if the transformations
$\gamma_{t}$ are $\mathcal{F}_{X}$-measurable and
$P_{X}(\gamma_{t}\Lambda)=P_{X}(\Lambda),t\in T,\Lambda\in\mathcal{F}_{X}$. If
the random field $X$ is generated by a measure preserving group of
transformations of $\Omega_{X}$, then, by Property (1), it is homogeneous.
Denote by $\mathcal{I}_{X}$ the $\sigma$-field of all events
$\Lambda\in\mathcal{F}_{X}$, which are invariant mod $P_{X}$ with respect to
all transformations $\gamma_{t}$, i.e.
$P_{X}(\Lambda\triangle\gamma_{t}\Lambda)=0,t\in T$. The group $\gamma$ is
said to be _metrically transitive_ if
$\mathcal{I}_{X}=\\{\Lambda:\Lambda\in\mathcal{F}_{X},P_{X}(\Lambda)=0\ {\rm
or}\ 1\\}$. The field is _ergodic_ if it is generated by a metrically
transitive measure preserving group $\gamma$.

Without loss of generality, we assume that
$(\Omega_{X},\mathcal{F}_{X},P_{X})$ is the probability space of function
type, i.e., $\Omega_{X}$ is the space of $\mathbb{R}^{d}$-valued ”sample
functions” $x(\cdot)$ on $T$, $\mathcal{F}_{X}$ is the $\sigma$-field
generated by the sets
$\\{x(\cdot):x(t)\in A\\},\;t\in T,A\in\mathcal{B}(\mathbb{R}^{d}),$ and
$P_{X}$ is a probability measure on $\mathcal{F}_{X}$. $X$ is the coordinate
random field: $X(t,x(\cdot))=x(t),$ where $t\in T,\,x(\cdot)\in\Omega_{X}$
(sometimes, we w rite $\omega$ instead of $x(\cdot)$, when we refer to an
element of $\Omega_{X}$).

Consider the invertible measurable ”shift” transformations $\gamma_{t},\;t\in
T,$ of $\Omega_{X},$ defined as follows : $\gamma_{t}x(\cdot)=x(\cdot+t)$; it
is clear that the family $\\{\gamma_{t}\\}$ is a group; moreover, $X$ is
generated by this group:

$X(0,\gamma_{t}x(\cdot))=X(0,x(\cdot+t))=x(0+t)=x(t)=X(t,\cdot)).$

The field is homogeneous, if and only if the shift transformations preserve
the measure $P_{X}$.

Each component field $X^{l}$ may be considered over the probability space
$(\Omega_{X}^{l},\mathcal{F}_{X}^{l},P_{X}^{l})$ where $\Omega_{X}^{l}$ is the
set of scalar functions $x^{l}(\cdot)$ on $T$, $\mathcal{F}_{X}^{l}$ is the
$\sigma$-field generated by the events $\\{x^{l}(\cdot):x^{l}(t)\in
D\\},\;t\in T,\;D\in\mathcal{B}(\mathbb{R})$. Let $A^{l}$ be an event in
$\mathcal{F}_{X}^{l}$; the event $\Lambda_{A^{l}}:=\\{x(\cdot):x^{l}(\cdot)\in
A^{l}\\}\in\mathcal{F}_{X}$, and $P_{X}^{l}$ is the projection of the measure
$P_{X}$ onto $\mathcal{F}^{l}$: for each $A^{l}\in\mathcal{F}_{X}^{l},\
P_{X}^{l}(A^{l})=P_{X}(\Lambda_{A^{l}})$. $X^{l}$ is the coordinate field:
$X^{l}(t,x^{l}(\cdot))=x^{l}(t),$ where $t\in T,$
$x^{l}(\cdot)\in\Omega_{X}^{l}$.

In what follows _we assume that the fields $X^{l}$ are homogeneous_, i.e., for
each $l$ all transformations $\gamma_{t}^{l}$ preserve the measure
$P_{X}^{l}$. It is also assumed that $E_{X}[|X^{l}(0)|]<\infty$ ($l=1,...,d$)
then we set $\mu^{l}:=E_{X}[X^{l}(0)].$ If $E_{X}[|X^{l}(0)|^{2}]<\infty,$
then the variances are denote by $\sigma^{l}:=Var_{X}[X^{l}(0)].$

_If $T=\mathbb{R}^{m}$, we always assume that the random fields
$X^{l}(t)=X^{l}(t,x^{l})$ are $\mathcal{B}\times\mathcal{F}^{l}$-measurable_;
by the Fubini theorem, these assumptions imply that _the sample functions
$x^{l}(\cdot)$ are Borel measurable with $P_{X}^{l}$-probability 1 and for
each finite Borel measure $Q$ the integral $\int_{T}x^{l}(t)Q(dt)$ exists with
$P_{X}^{l}$-probability 1_.

Denote by $\mathcal{I}^{l}_{X}$ the $\sigma$-field of all $\gamma$-invariant
$\mod(P^{l}_{X})$ events in $\mathcal{F}_{X}^{l}$ (we remind that $X^{l}$ is
said to be ergodic if $\mathcal{I}^{l}_{X}$ is trivial).
$E_{X}[X^{l}(0)|\mathcal{I}^{l}_{X}],\linebreak
Var_{X}[X^{l}(0)|\mathcal{I}^{l}_{X}]$ are the conditional expectation and
variance; _in the sequel it is assumed that for each $l$
$Var_{X}[X^{l}(0)|\mathcal{I}_{X}]>0$ with $P^{l}_{X}$-probability 1_.

In some cases it is assumed that the $\mathbb{R}^{d}$-valued random field $X$
is homogeneous, that is the transformations $\gamma_{t}$ preserve the measure
$P_{X}$; then the transformations $\gamma_{t}^{l}$ preserve the measures
$P_{X}^{l}$, and the components $X^{l}$ are also homogeneous; if the random
field $X$ is ergodic, then all its component fields $X^{l}$ are ergodic, too.

###  1.4 Randomizing random vectors

Let $\\{q_{n}^{l}\\}$ be sequences of probability measures on $\mathcal{B},\
(l=1,...,d),\;d\in\mathbb{N}$ (these sequences may coincide for some or even
all $l$). For each natural $n$ we consider $d$ mutually independent
$k_{n}$-dimensional random vectors
$\tau_{n}^{l}=(\tau^{l}_{n,1},...,\tau^{l}_{n,k_{n}}),\;$ $l=1,...,d,$ over a
probability space $(\Omega_{\tau},\mathcal{F}_{\tau},P_{\tau})$ possessing the
following properties:

a) the vectors $\tau_{n}^{l}$ do not depend on the random field $X,$

and

b) the components of each vector $\tau^{l}$ are i.i.d. $T$-valued random
vectors with the distribution $q_{n}^{l}.$

(Of course, if $T=\mathbb{Z}^{m}$ and if for some $l$ the support of
$q_{n}^{l}$ is finite, then some points may appear in the sample
$\tau^{l}_{n,1},...,\tau^{l}_{n,k_{n}}$ several times). It is assumed that
$k_{n}\uparrow\infty$ as $n\to\infty$ (the sequence $\\{k_{n}\\}$ may also
depend on $l$; to simplify the notation, we always drop the index $l$ in
$k_{n}^{l}$). Since each scalar field
$X^{l}=X^{l}(t,\omega)$ is a $\mathcal{B}\times\mathcal{F}_{X}$-measurable
function, $X(\tau^{l}_{n,i},\omega)$ are random variables over the probability
space

$(\Omega_{X,\tau},\mathcal{F}_{X,\tau},P_{X,\tau}):=(\Omega_{X}\times\Omega_{\tau},\mathcal{F}_{X}\times\mathcal{F}_{\tau},P_{X}\times
P_{\tau})$

.

###  1.5  Notation

In the sequel we also use the following notation:

$\bullet$ $E_{X}$, $E_{\tau}$ and $E_{X,\tau}$ denote the expectation with
respect to the measure $P_{X}$, $P_{\tau}$ and $P_{X,\tau}.$

$\bullet$ $Y_{n}\to Y$ $P$-a.s., $Y_{n}\overset{P}{\to}Y$ mean convergence
almost sure, respectively in probability.

$\bullet$ $Y_{n}\overset{Q}{\Longrightarrow}Y$ means convergence in
distribution with respect to the measure $Q.$

$\bullet$ $a:=b$ means that the quantity $a$ is defined by the expression $b$.

$\bullet$ $\mathbb{R},\mathbb{Z},\mathbb{N}$ denote the sets of real numbers,
of integers and of natural numbers.

$\bullet$ $T=\mathbb{R}^{m}$ or $\mathbb{Z}^{m}\ (m\in\mathbb{N})$.

$\bullet$ $\mathbbm{1}_{A}$ or $Ind(A)$ denotes the indicator of the set $A.$

###  1.6 Terminology

All results are stated in the terms of homogeneous random fields (of course,
they are valid for stationary random sequences and processes, too). If $m=1$,
the words ”homogeneous random field” mean ”stationary random process” or
”stationary random sequence”; the words ”ball”, ”cube” and ”parallelepiped”
mean ”interval” and, if $m=2$, these words mean ”sphere” ”square” and
”parallelogram, respectively.

##  2 Randomization using uniform distributions on
subsets $T_{n}^{l}$ of $T$

Let $X(t)=(X^{l}(t),...,.X^{d}(t)),\;t\in T,$ be a homogeneous random field on
$T$, $\\{T_{n}^{l}\\}$ be sequences of bounded Borel sets of positive measure
in $T\\\ (l=1,...,d),\;d\in\mathbb{N}$ (these sequences may coincide for some
or even all $l$). It is supposed that $\lambda(T^{l}_{n})\to\infty$ as
$n\to\infty.$

We consider the $T_{n}$-valued random vectors
$\tau^{l}_{n,1},...,\tau^{l}_{n,k_{n}},\;\;l=1,...,d,$ $n\in\mathbb{N}$,
introduced in Subsect. 1.4.

######  Lemma 1.

If $f$ is a measurable function on $\mathbb{R}$ such that
$E_{X}[|f(X(0))|]<\infty$, then $E_{X,\tau}[|f(X(\tau^{l}_{n,i}))|]<\infty$
and with $P_{X}$-probability 1
$E_{\tau}[|f(X(\tau^{l}_{n,i}))|]<\infty$.

###### Proof.

It follows immediately from the Fubini - Tonelli Theorem. ∎

Assume that for each $l\in\\{1,...,d\\}$ and for each natural number $n$ the
random vectors $\tau^{l}_{n,i},\;i=1....,k_{n}$, introduced in Subsect. 1.4,
are uniformly distributed (with respect to $\lambda$) on the set $T_{n}^{l}$.
In this section we use the following estimators of $\mu^{l}$ and
$(\sigma^{l})^{2}$:

$\displaystyle
M_{n}^{l}:=\frac{1}{\lambda(T_{n}^{l})}\int_{T_{n}^{l}}X^{l}(t)\lambda(dt),$
$\displaystyle
V_{n}^{l}:=\frac{1}{\lambda(T_{n}^{l})}\int_{T_{n}^{l}}(X^{l}(t)-M_{n}^{l})^{2}\lambda(dt)=\frac{1}{\lambda(T_{n}^{l})}\int_{T_{n}^{l}}(X^{l}(t))^{2}\lambda(dt)-(M_{n}^{l})^{2}.$

If $\omega\in\Omega_{X}$ is fixed, the random variables
$X^{l}(\tau^{l}_{n,i},\omega)$ form $d$ triangular arrays and for each $n$
they are independent and identically distributed. For each measurable function
$f$ on $\mathbb{R}$, such that $f[X^{l}(t)]$ is integrable on $T_{n}^{l}$,

$E_{\tau}[f(X^{l}(\tau^{l}_{n,i}))]=\frac{1}{\lambda(T_{n}^{l})}\int_{T_{n}^{l}}f[X^{l}(t)]\lambda(dt),\;\;i=1,...,k_{n},\;\;l=1,...,d.$

In particular,

$E_{\tau}[X^{l}(\tau^{l}_{n,i})]=M^{l}_{n};\;\;Var_{\tau}[X^{l}(\tau^{l}_{n,i})]=V^{l}_{n}.$

The above relation implies:

$\displaystyle
E_{\tau}[\sum_{i=1}^{k_{n}}f(X^{l}(\tau^{l}_{n,i}))]=k_{n}\frac{1}{\lambda(T_{n}^{l})}\int_{T_{n}^{l}}f[X^{l}(t)]\lambda(dt)\
(l=1,...,d).$ (2)

######  Definition 1.

We say that a sequence of Borel sets $\\{T_{n}\\}$ is _pointwise averaging_ if
the following _Pointwise Ergodic Theorem_ (PET) holds with this sequence:

_Let $f(\cdot)$ be a measurable function on $\mathbb{R}$. If $X$ is a scalar
homogeneous random field on $T$ over $(\Omega_{X},\mathcal{F}_{X},P_{X})$ with
$E_{X}[|f(X(0))|]<\infty$ and $\mathcal{I}_{X}$ is the $\sigma$-field of
shift-invariant $\mod(P_{X})$ events in $\mathcal{F}_{X}$, then with
$P_{X}$-probability 1_

$\lim_{n\to\infty}\frac{1}{\lambda(T_{n})}\int_{T_{n}}f(X(t))\lambda(dt)=E_{X}[f(X(0))|\mathcal{I}_{X}].$

######  Example 1.

In $\mathbb{R}^{m}$ each increasing sequence of bounded convex sets
$T_{n}^{l}\in\mathbb{R}^{m}$, containing balls $B_{n}$ of radii
$r(B_{n})\to\infty$, is pointwise averaging (see Corollary 3.3 in Ch. 6 in
[64] ). In particular, any sequence of concentric balls $B_{n}$ with
$r(B_{n})\to\infty$ and any increasing sequence of parallelepipeds
$T_{n}\subset\mathbb{R}^{m}$ with infinitely growing edges is pointwise
averaging (in particular, the sequence of cubes $[0,n]^{m}$ has this
property). The intersections of the mentioned sets with $\mathbb{Z}^{m}$ form
pointwise averaging sequences in $\mathbb{Z}^{m}$ (see Subsect. 5.2 in [61]).

######  Example 2.

Let $A$ be a compact set in $\mathbb{R}^{m}$, with $\lambda(A)>0$ and star-
shaped with respect to $0$. If $s_{n}\uparrow\infty$, the sequence of
homothetic sets $\\{s_{n}A\\}$ is pointwise averaging (see Example 2.9 in Ch.
5 in [64] or Subsection 5.4.1 in [61]).

######  Remark 1.

Along with the pointwise averaging sequences of sets, _mean averaging_
sequences may be considered, i.e. sequences $T_{n}$ for which The _Mean
Ergodic Theorem_ (MET) for each homogeneous random field with
$E[(X(0))^{2}]<\infty$:

$E_{X}\left[|\frac{1}{\lambda(T_{n})}\int_{T_{n}}X(t)\lambda(dt)-E_{X}[X(0)|\emph{I}_{X}]|^{2}\right]\to
0.$

Under our condition $E[|X(0)|^{2}]<\infty,$ each pointwise averaging sequence
is mean averaging (see Corollary 2 in Subsect 9.4 in [45]). Consider the
Hilbert subspace $H_{X}$ of $L^{2}(\Omega_{X},\mathcal{F}_{X},P_{X})$, spanned
by the random variables $X(t),$
$t\in T$; it is invariant with respect to the shift transformations:
$W(\gamma_{t}\omega)\in H_{X}$ if $W(\omega)\in H_{X}$; denote by $I$ the
subspace of all random variables invariant with respect to all $\gamma_{t}$.
The conditional expectation $E_{X}[X(0)|\emph{I}_{X}]=\tilde{E}_{X}[X(0)|I]$,
the orthogonal projection of $X(0)$ onto $I$. This random variable equals
$E[X(0)]$, if $X$ is ”wide-sense ergodic”, i.e., if $I=\mathbb{R}$; of course
this condition is much weaker than ergodicity, which assures that the subspace
of all $\gamma$-invariant random variables in
$L^{2}(\Omega_{X},\mathcal{F}_{X},P_{X})$ coincides with $\mathbb{R}$. The
class of the mean averaging sequences is rather wide; in particular, the
monotonicity condition in Example 1 is redundant. The MET is essentially used
in the sequel (see also Remark 8 where the speed of convergence in the MET is
discussed).

##  3 Randomized Lindeberg condition

Let us remind that the main probability space has the following form
$(\Omega_{X,\tau},\mathcal{F}_{X,\tau},P_{X,\tau})=(\Omega_{X}\times\Omega_{\tau},\mathcal{F}_{X}\times\mathcal{F}_{\tau},P_{X}\times
P_{\tau})$.

######  Proposition 1.

Let $X$ be a homogeneous random field, $E[(X(0))^{2}]<\infty$, and let
$\\{T_{n}\\}$ be a pointwise averaging sequence of sets. Then for
$P_{X}$-almost all $\omega\in\Omega_{X}$ the Lindeberg condition is fulfilled:

for each $\varepsilon$ as $n\to\infty$

$\displaystyle
L_{n}(\omega):=\frac{\sum_{i=1}^{k_{n}}E_{\tau}[[X(\tau_{n,i},\omega)-M_{n}(\omega)]^{2}{\mathbf{1}}_{B_{n}}]}{{k_{n}}V_{n}(\omega)}\to
0,$

where $B_{n}=\\{|X(\tau_{n,i},\omega)-M_{n}(\omega)|>\varepsilon
k_{n}^{\frac{1}{2}}(V_{n})^{\frac{1}{2}}\\}.$

###### Proof.

For a fixed ”good” $\omega$, we will apply the Lindeberg theorem (see, e.g.,
Theorem 27.2 in [3]) to the random variables over the space
$(\Omega_{\tau},\mathcal{F}_{\tau},P_{\tau})$. By relation (2), for each
$\omega\in\Omega_{X}$ and for each $\varepsilon>0$, the Lindeberg fraction for
the random variables $X(\tau_{n,i}(w),\omega),\;i=1,...,k_{n},$ is given by

$\displaystyle
L_{n}(\omega)=\frac{1}{V_{n}(\omega)}\frac{1}{\lambda(T_{n})}\int_{T_{n}}(X(t,\omega)-M_{n}(\omega))^{2}{\mathbf{1}}_{C_{n}(t)}\lambda(dt),$

where $C_{n}(t)=\\{|X(t,\omega)-M_{n}(\omega)|>\varepsilon
k_{n}^{\frac{1}{2}}(V_{n})^{\frac{1}{2}}\\}.$

We have to prove that with $P_{X}$-probability 1 $L_{n}(\omega)\to 0$. Since
the sequence $\\{T_{n}\\}$ is pointwise averaging, by the PET, with
$P_{X}$-probability 1

$M_{n}(\omega)\to z(\omega),$ (3) $\displaystyle
V_{n}(\omega)=\frac{1}{\lambda(T_{n})}\int_{T_{n}}(X(t,\omega)-M_{n}(\omega))^{2}\lambda(dt)\to
v(\omega),$ (4)

where

$\displaystyle z(\omega)=E_{X}[X(0,\omega)|\mathcal{I}_{X}](\omega),$ (5)
$\displaystyle
v(\omega)=Var_{X}[(X(0),\omega)^{2}|\mathcal{I}_{X}](\omega)>0.$ (6)

It is clear that

$\displaystyle L_{n}(\omega)\leq\Sigma^{(1)}_{n}+\Sigma^{(2)}_{n},$

where

$\Sigma^{(1)}_{n}=\frac{1}{V_{n}(\omega)}\frac{2}{\lambda(T_{n})}\int_{T_{n}}(X(t,\omega)-z(\omega))^{2}{\mathbf{1}}_{C_{n}(t)}\lambda(dt),$
$\Sigma^{(2)}_{n}=\frac{1}{V_{n}(\omega)}\frac{2}{\lambda(T_{n})}\int_{T_{n}}(z(\omega)-M_{n})^{2}{\mathbf{1}}_{C_{n}(t)}\lambda(dt).$

Due to (3) with $P_{X}$-probability 1

$\Sigma^{(2)}_{n}\leq(z(\omega)-M_{n}(\omega))^{2}\frac{2}{V_{n}(\omega)}\to
0.$ (7)

Consider $\Sigma^{(1)}_{n}.$ Denote by $\Lambda\in\mathcal{F}_{X}$ the set of
$P_{X}$-measure 1 in $\Omega$ where the limits (3) and (4) exist, and fix some
$\omega\in\Lambda$. It is clear that $k_{n}V_{n}(\omega)\to\infty$ as
$n\to\infty$. For each $C>0$, let us also fix some
$m_{C}(\omega)\in\mathbb{N}$ such that, if $n>m_{C}(\omega)$, then
$|M_{n}(\omega)-z(\omega)|<\frac{1}{2}\varepsilon C$ and
$(k_{n}V_{n}(\omega))^{\frac{1}{2}}>C>0$.

As for $n>m_{C}(\omega)$

$\\{|X(t,\omega)-M_{n}|>\varepsilon
k_{n}^{\frac{1}{2}}(V_{n})^{\frac{1}{2}}\\}\subset\\{|X(t,\omega)-z(\omega)|>\frac{\varepsilon
C}{2}\\}\cup\\{|z(\omega)-M_{n}|>\frac{\varepsilon C}{2}\\},$

we have:

$\Sigma^{(1)}_{n}\leq\delta^{(1)}_{n}+\delta^{(2)}_{n},$ (8)

where

$\delta^{(1)}_{n}=\frac{1}{V_{n}(\omega)}\frac{2}{\lambda(T_{n})}\int_{T_{n}}(X(t,\omega)-z(\omega))^{2}{\mathbf{1}}_{\\{|X(t,\omega)-z(\omega)|>\frac{\varepsilon
C}{2}\\}}\lambda(dt),$
$\delta^{(2)}_{n}=\frac{1}{V_{n}(\omega)}\frac{2}{\lambda(T_{n})}\int_{T_{n}}(X(t,\omega)-z(\omega))^{2}{\mathbf{1}}_{\\{|z(\omega)-M_{n}|>\frac{\varepsilon
C}{2}\\}}\lambda(dt).$

By the PET, with $P_{X}$-probability 1

$\lim_{n}\delta^{(1)}_{n}=\frac{2}{v(\omega)}E\left[(X(0,\omega)-z(\omega))^{2}{\mathbf{1}}_{\\{|X(0,\omega)-z(\omega)|>\frac{\varepsilon
C}{2}\\}}\right],$ (9)

and due to (3)

$\lim_{n}\delta^{(1)}_{n}=\frac{2}{v(\omega)}\lim_{n}\frac{1}{\lambda(T_{n})}\int_{T_{n}}(X(t,\omega)-z(\omega))^{2}\lambda(dt)\cdot\lim_{n}{\mathbf{1}}_{\\{|z(\omega)-M_{n}|>\frac{\varepsilon
C}{2}\\}}=0.$ (10)

It follows from (7)-(10) that

$\limsup_{n}L_{n}(\omega)\leq
2E\left[(X(0,\omega)-z(\omega))^{2}{\mathbf{1}}_{\\{|X(0,\omega)-z(\omega)|>\frac{\varepsilon
C}{2}\\}}\right].$

Letting $C$ tend to infinity, we finally obtain with $P_{X}$-probability 1

$\lim_{n}L_{n}(\omega)=0.$

∎

##  4 Generalizations of the Central Limit Theorem

###  4.1 On the CLT

The Central Limit Theorem (CLT) is one of the remarkable statements of
Probability. Originally proved for sequences of independent random variables,
in many works this theorem was generalized to stationary random processes and
homogeneous random fields under some additional assumptions: Markov, strong
mixing and/or rather strong moment conditions, etc. (see, e.g., [4, 7, 8, 10,
11, 12, 15, 19, 21, 31, 32, 34, 43, 46, 47, 50, 51, 53], [63]-[68]).

However, the CLT may fail even when the sequence $X_{1},X_{2},...,$ is
stationary, quite strongly mixing, orthogonal and with
$\lim\frac{1}{n}\text{Var}(\sum_{i=1}^{n}X_{i})=\sigma^{2}>0$ [28]; two other
interesting counterexamples are provided in [8]. Earlier Chung [13] and
Davydov [16, 17] proved that the CLT may fail for strictly stationary
irreducible aperiodic Markov chains with a countable set of states (these and
other counterexamples can be found in vol. 3, Chapters 30 and 31 in the book
[8]). Ibragimov and Linnik [33] have constructed a strongly mixing stationary
sequence of random variables $X_{1},X_{2},...,$ with finite variances such
that the self-normalized sums
$(Var[\sum_{i=1}^{n}X_{i}])^{-\frac{1}{2}}\sum_{i=1}^{n}X_{i}$ converge in
distribution to a non-normal random variable (see Ch. 19, §5 therein). A
rather common case when the CLT fails is when
$\lim\frac{1}{n}\text{Var}[\sum\limits_{i=1}^{n}{X_{i}}]=0$ (see, e.g., [20],
p.153).

A well-known source of counterexamples for the classical CLT for strict sense
stationary random sequences is the class of coboundaries with respect to
measure preserving transformations. Let $\gamma:\omega\mapsto\gamma\omega$ be
an ergodic measure preserving transformation of a probability space
$(\Omega,\mathcal{F},P)$; let $\alpha\geq 1$; then for each $f\in
L^{\alpha}(\Omega,\mathcal{F},P),f\neq 0$ a.s., the random sequence
$X_{i}^{(f)}=f(\gamma^{i}\omega),i=1,2,...,$ is an ergodic strict sense
stationary sequence and $E[|X_{1}^{(f)}|^{\alpha}]<\infty$ (see, e.g., [41],
§1.4). Let $g\in L^{\alpha}(\Omega,\mathcal{F},P)$ and let $f$ be the
_coboundary_ of $g$: $f(\omega)=g(\gamma\omega)-g(\omega)$. It is clear that
$E(X_{i}^{(f)})=0,$ $i=1,2,...$. We have also:
$\sum_{i=1}^{n}X_{i}^{(f)}=\sum_{i=1}^{n}f(\gamma^{i}\omega)=g(\gamma^{n+1}\omega)-g(\gamma\omega)$,
hence $E[|\sum_{i=1}^{n}X_{i}^{(f)}|^{\alpha}\leq
2^{\alpha}||g||_{L^{\alpha}}^{\alpha}]$, and
$E[|n^{-\beta}\sum_{i=1}^{n}X_{i}^{(f)})|^{\alpha}]\to 0$ if $\beta>0$ (when
$\alpha>1$, the converse is true: if $E[f]=0$ and
$\sup_{n}E[(\sum_{i=1}^{n}X_{i}^{(f)})^{\alpha}]<\infty$, it is a coboundary
of some $g\in L^{\alpha}$; see Lemma 5 in [9]); the set of coboundaries is
dense in the space $L^{\alpha}(\Omega,\mathcal{F},P)\ominus\mathbb{R}$, by the
ergodic decomposition. It is evident that, if $f$ is a coboundary of a
function $g\in L^{\alpha}$, then $n^{-\beta}\sum_{i=1}^{n}X_{i}^{(f)}\to 0$ in
probability for any $\beta>0$, and, if $g$ is bounded, then this is true in
the sense of a.s. convergence.

There is an extensive literature related to limit theorems for “long memory”
(or “long-range dependent”) stationary sequences, where the slowly decreasing
dependence between receding terms is characterized by the rate of decrease of
the correlation function or by properties of the spectrum near 0) - see [1],
[26], [56] and the references therein.

###  4.2 Main results of this section

In the previous work by the second author [62], CLTs for homogeneous random
fields on $\mathbb{R}^{m}$ and $\mathbb{Z}^{m}\ (m\geq 1)$, satisfying the
condition: $E[|X(0)|^{2+\delta}]<\infty$ have been proved. In the present
article the authors prove that these theorems are valid under the weaker
condition $E[(X(0))^{2}]<\infty$.

In this paper we present randomized versions of the CLT, which are valid for
each ergodic homogeneous measurable random field $X(\cdot)$ on
$\mathbb{R}^{m}$ or
$\mathbb{Z}^{m}\ (m\geq 1)$, (in particular, for each ergodic stationary
random process and each ergodic stationary random sequence) with a finite
second moment (in some versions ergodicity may be omitted). Specifically,
observations of the random field at randomly chosen points are used.

If the field X is multivariate and each of its components satisfies the above
conditions, another feature is obtained by the randomization: in Theorems 1-29
and Corollaries 1, 2 the components of the limit normal vector are independent
whatever are the components of $X$. _When $E[|X(0)|^{2}]<\infty$, these
theorems are valid also in all cases, mentioned in Subsection 1.1, when the
conventional CLT fails._

For example, consider the stationary random sequences $\\{X_{i}^{(f)}\\}$,
which have been introduced in Subsection 4.1 (each sequence is specified by
some coboundary $f\in L^{2}(\Omega,\mathcal{F},P))$; we have:
$E[|X_{1}^{(f)}|^{2}]<\infty$; as mentioned above, the set of coboundaries is
dense in $L^{2}(\Omega,\mathcal{F},P)\ominus\mathbb{R}$, and the conventional
CLT fails for $\\{X_{i}^{(f)}\\}$, if $f$ a coboundary: the limit is
degenerate ($\delta_{0}$) and not normal. However, the randomized CLTs are
valid for all sequences $\\{X_{i}^{(f)}\\}$ with $E[|X_{i}^{(f)}|^{2}<\infty$.

Randomization allowed us to assume that the field is only homogeneous (in some
statements it is assumed that the field is also ergodic) and its second moment
is finite.

The main tools, used in the proofs, are the Lindeberg CLT and the Pointwise
and Mean Ergodic Theorems.

To illustrate our results, we state a simple corollary of Theorem 3. _Let
$X(t),t\in\mathbb{R}$, be an ergodic stationary measurable random process and
$E[(X(0))^{2}]<\infty$; denote
$\sigma^{2}=Var[X(0)],M_{n}=\frac{1}{n}\int_{0}^{n}X(t)dt$. Let $\tau_{n,i}\\\
(n=1,2,...$, and $i=1,...,n)$ be random variables, independent of $X$, and,
for each $n$, independent of each other and uniformly distributed on $[0,n]$.
Then, if $n\to\infty$,_

$\frac{\sum_{i=1}^{n}(X(\tau_{n,i})-M_{n})}{\sigma\sqrt{n}}\Longrightarrow
N(0,1).$

A slightly more complex corollary with $\mu=E[X(0)]$ instead of $M_{n}$ can be
derived from Theorem 4; as usual, if $\sigma$ is known, this statement may be
used for consistent statistical inference on the expectation $\mu$.

###  4.3 Randomized Central Limit Theorem

######  Assumptions 1.

In Theorems 1–5, all $X^{l},\;l=1,...,d,$ are homogeneous random fields and,
for all $l$,

$E_{X}[|X^{l}(0)|^{2}]<\infty.$ (11)

If $X^{l}$ is not ergodic, we assume $Var_{X}[X^{l}(0)|\mathcal{I}^{l}_{X}]>0$
with $P^{l}_{X}$-probability 1, $l=1,...,d$.

We remind that $k_{n}\in\mathbb{N},\;k_{n}\uparrow\infty$.

For each $l$, $\\{T_{n}^{l}\\}$ is a pointwise averaging sequence of sets.

######  Remark 2.

We will prove that, together with the conditions mentioned above, in the non-
ergodic case, the condition $Var_{X}[X^{l}(0)|\mathcal{I}^{l}_{X}]>0\
P^{l}_{X}{\rm-a.s.},\\\ l=1,...,d,$ is _sufficient_ for the relation (13) to
hold with $P_{X}$-probability 1. And, for each fixed $l$, it is also
_necessary_ for this statement to hold for the component $X^{l}$. Indeed, if
the variance of the conditional distribution
$Var_{X}[X^{l}(0)|\mathcal{I}^{l}_{X}]=0$, then
$X^{l}(0)=E_{X}[X^{l}(0)|\mathcal{I}^{l}_{X}]$ $P_{X}^{l}$-a.s., hence
$X^{l}(0)$ is $\mathcal{I}^{I}_{X}$-measurable and
$X^{l}(t,\omega^{l})=X^{l}(0,\gamma_{t}^{l}\omega^{l})=X^{l}(0,\omega^{l})$
$P_{X}^{l}$-a.s. so, for each $t\in T$, $X^{l}(t)=X^{l}(0)$ $P_{X}^{l}$-a.s.
It is clear, that this dependence is too strong for any version of the CLT to
hold for $X^{l}$. In this case, it becomes senseless: note that
$M^{l}_{n}=X^{l}(0),V_{n}^{l}=0;$ therefore, the expression

$\frac{\sum_{i=1}^{k_{n}}(X^{l}(\tau^{l}_{n,i})-M_{n}^{l})}{k_{n}^{\frac{1}{2}}(V^{l}_{n})^{\frac{1}{2}}}=\frac{{k_{n}}(X^{l}(0)-M^{l}_{n})}{k_{n}^{\frac{1}{2}}(V^{l}_{n})^{\frac{1}{2}}}$

(see (13) ) is of type $\frac{0}{0}$; conditions (17) and (28) for $X^{l}$ do
not hold, and the expressions (3), (4) for $X^{l}$ are of type $\frac{1}{0}$.

Let us remind that the probability space

$(\Omega_{X,\tau},\mathcal{F}_{X,\tau},P_{X,\tau})=(\Omega_{X}\times\Omega_{\tau},\mathcal{F}_{X}\times\mathcal{F}_{\tau},P_{X}\times
P_{\tau}).$

######  Lemma 2.

1\. Let $A_{n}(z)$ be a sequence of events in $\Omega_{X,\tau}$ depending on
$z\in\mathbb{R}^{d}$; for each $\omega\in\Omega_{X}$ denote:
$A_{n}(\omega,z)=\\{t\in\Omega_{\tau}:(\omega,t)\in A_{n}(z)\\}$, the
$\omega$-section of $A(z)$; if for each $z\in\mathbb{R}^{d}$ the limit

$\lim_{n\to\infty}P_{\tau}(A_{n}(\omega,z))=\psi(z)$

exists for $P_{X}$-almost all $\omega$, then for each $z\in\mathbb{R}^{d}$

$\lim_{n\to\infty}P_{X,\tau}(A_{n}(z))=\psi(z).$ (12)

2\. Let $\\{f_{n}(X,\tau)=(f_{n}^{1}(X,\tau),...,f_{n}^{d}(X,\tau)),n\in N\\}$
be a sequence of random vectors over $\Omega_{X,\tau}$ and let
$Z=(Z^{1},...,Z^{d})$ be a random vector over the same probability space. If
$f_{n}(X,\tau)\to Z$ in $P_{\tau}$-distribution with $P_{X}$-probability 1,
then $f_{n}(X,\tau)\to Z$ in $P_{X,\tau}$-distribution.

###### Proof.

1\. By the Fubini theorem, $A_{n}(\omega,z)\subset\mathcal{F}_{\tau}$ for
$P_{X}$-almost all $\omega\in\Omega_{X}$. By virtue of the Fubini theorem and
the Lebesgue Dominated Convergence theorem, we have for each
$z\in\mathbb{R}^{d}$:

$\displaystyle\lim_{n\to\infty}P_{X,\tau}(A_{n}(z))=\lim_{n\to\infty}E_{X,\tau}[1_{A_{n}(z)}]=\lim_{n\to\infty}E_{X}[E_{\tau}[1_{A_{n}(z)}]]=$
$\displaystyle\lim_{n\to\infty}E_{X}[P_{\tau}(A_{n}(\omega,z)]=E_{X}[\lim_{n\to\infty}P_{\tau}(A_{n}(\omega,z)]=E_{X}\psi(z)=\psi(z).$

2\. Apply Statement 1 with

$\displaystyle
A_{n}(z):=\\{(\omega,t)\in\Omega_{X,\tau}:f_{n}^{1}(\omega,t)\leq
z^{1},...,f_{n}^{d}(\omega,t)\leq z^{d}\\},$
$\displaystyle\psi(z)=P_{X}(Z^{1}\leq z^{1},...,Z^{d}\leq z^{d}).$

∎

We denote: $Z=(Z^{1},...,Z^{d})$ where $Z^{1},...,Z^{d}$ are independent
standard normal random variables.

######  Theorem 1.

Under Assumptions 1, with $P_{X}$-probability 1

$\left(\frac{\sum_{i=1}^{k_{n}}(X^{1}(\tau^{1}_{n,i})-M_{n}^{1})}{k_{n}^{\frac{1}{2}}(V^{1}_{n})^{\frac{1}{2}}},...,\frac{\sum_{i=1}^{k_{n}}(X^{d}(\tau^{d}_{n,i})-M_{n}^{d})}{k_{n}^{\frac{1}{2}}(V^{d}_{n})^{\frac{1}{2}}}\right)\overset{P_{\tau}}{\Longrightarrow}Z;$
(13)

if all $X^{l}$ are ergodic, then with $P_{X}$-probability 1

$\left(\frac{\sum_{i=1}^{k_{n}}(X^{1}(\tau^{1}_{n,i})-M_{n}^{1})}{k_{n}^{\frac{1}{2}}\sigma^{1}},...,\frac{\sum_{i=1}^{k_{n}}(X^{d}(\tau^{d}_{n,i})-M_{n}^{d})}{k_{n}^{\frac{1}{2}}\sigma^{d}}\right)\overset{P_{\tau}}{\Longrightarrow}Z.$
(14)

###### Proof.

We may apply the Lindeberg theorem and obtain: with $P_{X}$-probability 1 for
each $l$

$\displaystyle\frac{\sum_{i=1}^{k_{n}}(X^{l}(\tau^{l}_{n,i})-M^{l}_{n})}{k_{n}^{\frac{1}{2}}(V^{l}_{n})^{\frac{1}{2}}}\overset{P_{\tau}}{\Longrightarrow}Z^{l}.$
(15)

If $X$ is ergodic, then $(V^{l}_{n})^{\frac{1}{2}}\to\sigma^{l}$ with
$P_{X}$-probability 1, and, by the Slutsky theorem ([42], Theorem 23.3), with
$P_{X}$-probability 1

$\displaystyle\frac{\sum_{i=1}^{k_{n}}(X^{l}(\tau_{n,i})-M^{l}_{n})}{k_{n}^{\frac{1}{2}}\sigma^{l}}\overset{P_{\tau}}{\Longrightarrow}Z^{l}.$
(16)

Since for each $n$ the $d$ vectors
$(\tau^{l}_{n,i},i=1,...,k_{n}),\;\;l=1,...,d,$ are mutually independent,
using (15) and (16) we obtain the convergence (13), respectively (14), of
vectors. ∎

Since the limiting distributions in (13) and (14) with $P_{X}$-probability 1
are the same, we immediately deduce from the previous theorem the non-
conditional weak convergence:

######  Theorem 2.

Under Assumptions 1,

$\left(\frac{\sum_{i=1}^{k_{n}}(X^{1}(\tau^{1}_{n,i})-M_{n}^{1})}{k_{n}^{\frac{1}{2}}(V^{1}_{n})^{\frac{1}{2}}},...,\frac{\sum_{i=1}^{k_{n}}(X^{d}(\tau^{d}_{n,i})-M_{n}^{d})}{k_{n}^{\frac{1}{2}}(V^{d}_{n})^{\frac{1}{2}}}\right)\overset{P_{X,\tau}}{\Longrightarrow}Z;$

if all $X^{l}$ are ergodic, then with $P_{X}$-probability 1

$\left(\frac{\sum_{i=1}^{k_{n}}(X^{1}(\tau^{1}_{n,i})-M_{n}^{1}}{k_{n}^{\frac{1}{2}}\sigma^{1}},...,\frac{\sum_{i=1}^{k_{n}}(X^{d}(\tau^{d}_{n,i})-M_{n}^{d})}{k_{n}^{\frac{1}{2}}\sigma^{d}}\right)\overset{P_{X,\tau}}{\Longrightarrow}Z.$

The following Theorems 3 and 4 can be deduced subsequently from Theorems 1-2
literarily as Th. 3 and Th. 4 in [62].

######  Theorem 3.

In addition to the above Assumptions 1, assume that

$k_{n}^{\frac{1}{2}}(M^{l}_{n}-\mu^{l})\to 0\ \text{with $P_{X}$-probability
1},\ l=1,...,d^{)}.$ (17)

Then with $P_{X}$-probability 1

$\left(\frac{\sum_{i=1}^{k_{n}}(X^{1}(\tau^{1}_{n,i})-\mu^{1})}{k_{n}^{\frac{1}{2}}(V^{1}_{n})^{\frac{1}{2}}},...,\frac{\sum_{i=1}^{k_{n}}(X^{d}(\tau^{d}_{n,i})-\mu^{d})}{k_{n}^{\frac{1}{2}}(V^{d}_{n})^{\frac{1}{2}}}\right)\overset{P_{\tau}}{\Longrightarrow}Z;$

if all $X^{l}$ are ergodic, then with $P_{X}$-probability 1

$\left(\frac{\sum_{i=1}^{k_{n}}(X^{1}(\tau^{1}_{n,i})-\mu^{1})}{k_{n}^{\frac{1}{2}}\sigma^{1}},...,\frac{\sum_{i=1}^{k_{n}}(X^{d}(\tau^{d}_{n,i})-\mu^{d})}{k_{n}^{\frac{1}{2}}\sigma^{d}}\right)\overset{P_{\tau}}{\Longrightarrow}Z.$

######  Theorem 4.

Let the conditions stated in Assumptions 1 be fulfilled and

$k_{n}^{\frac{1}{2}}(M^{l}_{n}-\mu^{l})\overset{P_{X}}{\Longrightarrow}0.$
(18)

Then

$\left(\frac{\sum_{i=1}^{k_{n}}(X^{1}(\tau^{1}_{n,i})-\mu^{1})}{k_{n}^{\frac{1}{2}}(V^{1}_{n})^{\frac{1}{2}}},...,\frac{\sum_{i=1}^{k_{n}}(X^{d}(\tau^{d}_{n,i})-\mu^{d})}{k_{n}^{\frac{1}{2}}(V^{d}_{n})^{\frac{1}{2}}}\right)\overset{P_{X,\tau}}{\Longrightarrow}Z;$

if all $X^{l}$ are ergodic, then

$\left(\frac{\sum_{i=1}^{k_{n}}(X^{1}(\tau^{1}_{n,i})-\mu^{1})}{k_{n}^{\frac{1}{2}}\sigma^{1}},...,\frac{\sum_{i=1}^{k_{n}}(X^{d}(\tau^{d}_{n,i})-\mu^{d})}{k_{n}^{\frac{1}{2}}\sigma^{d}}\right)\overset{P_{X,\tau}}{\Longrightarrow}Z.$

Condition (18) puts some restrictions to the growth of the number $k_{n}$ of
observations of the values of the field $X$. It is often of the form
$k_{n}=o(\lambda(T_{n}))$ (see Remark 8). On the other hand, the accuracy of
the approximation of the normal distribution by the CLT is higher when $k_{n}$
is large. An obvious way to increase $k_{n}$ without violation of this
condition is to increase $\lambda(T_{n})$ proportionally. Another way to
increase the number of observations without breaking this condition puts more
work for the statistician; it is based on a special case of Theorem 4, when,
instead of $X$, an auxiliary $\mathbb{R}^{wd}$-valued random field $Y$ is
considered, in which each component $X^{l}$ participates $w$ times.

######  Corollary 1.

Let the conditions of Theorem 4 be fulfilled. Consider the family of
independent random variables
$\\{\tau^{l,u}_{n,i},n\in\mathbb{N},\;\;1\leq i\leq k_{n},1\leq l\leq
d,\;1\leq u\leq w\\}$ which, for each $n,l,u,i,$ are distributed uniformly on
$T_{n}^{l}$. Then

$\frac{\sum_{i=1}^{k_{n}}(X^{l}(\tau^{l,u}_{n,i})-\mu^{l})}{k_{n}^{\frac{1}{2}}(V_{n}^{l})^{\frac{1}{2}}}\overset{P_{X,\tau}}{\Longrightarrow}Z^{l,u}\
,\;\;l=1,...d,\;\;u=1,...w.$

where all random variables $Z^{l,u},\;\;l=1,...,d,\;\;u=1,...,w,$ are standard
normal and independent.

If the field $X$ is ergodic, the ”empirical” standard deviations $V_{n}^{l}$
can be replaced by the ”theoretical” standard deviations $\sigma^{l}$.

###### Proof.

Apply Theorem 4 to the random fields $Y^{l,u}(t):=X^{l}(t),\;\;t\in
T,\;\;l=1,...,d,\;\;u=1,...,w.$ ∎

We consider a generalization of this corollary to various parameters of the
marginal or multidimensional distributions of the field $X$; for simplicity we
assume that $X$ is $\mathbb{R}$-valued. Let $\theta^{l},l=1.,,,,d,$ be
parameters of the distribution of a vector $(X(t_{1}),...,X(t_{k}))$. We
assume that there exists a $\mathcal{B}(\mathbb{R}^{k})$-measurable functions
$f^{l}(x_{1},...,x_{k})$ such that

$E_{X}[|f^{l}(X(t_{1}),...,X(t_{k}))|^{2}]<\infty,$

and $\theta^{l}=E_{X}[f^{l}(X(t_{1}),...,X(t_{k}))].$ Denote:

$\sigma_{f}^{l}:=[Var_{X}[f^{l}(X(t_{1}),...,X(t_{k}))]^{\frac{1}{2}},$
$M^{l}_{f,n}:=\frac{1}{\lambda(T_{n})}\int_{T_{n}}f^{l}(X(t_{1}+t),...,X(t_{k}+t))\lambda(dt),$
$V^{l}_{f,n}:=\frac{1}{\lambda(T_{n})}\int_{T_{n}}(f^{l}(X(t_{1}+t),...,X(t_{k}+t))-M^{l}_{f,n})^{2}\lambda(dt).$

For each $l$ we consider the random field

$Y_{f}^{l}(t):=f^{l}(X(t_{1}+t),...,X(t_{k}+t)),\;t\in T,$

over $(\Omega_{X},\mathcal{F}_{X},P_{X})$). Note that
$Y_{f}^{l}(0):=f^{l}(X(t_{1}),...,X(t_{k}+t))$, so _the condition
$E_{X}|Y_{f}^{l}(0)|^{2}<\infty$ is fulfilled_; the finite-dimensional
distributions of the field $Y_{f}^{l}$ coincide with finite-dimensional
distributions of $X^{l}$, and $Y_{f}^{l}$ is generated by the shift
transformations: by Property (1),

$\displaystyle
Y_{f}^{l}(0,\gamma_{t}\omega)=f^{l}(X(t_{1},\gamma_{t}\omega),...,X(t_{k},\gamma_{t}\omega))=$
$\displaystyle
f^{l}(X(t_{1}+t),\omega),...,X(t_{k}+t),\omega))=Y_{f}^{l}(t,\omega);$

therefore, the field $Y_{f}^{l}$ is homogeneous, and, if the field $X^{l}$ is
ergodic, the field $Y_{f}^{l}$, is ergodic, too.

For example, if $\theta$ is a mixed moment, i.e.

$\theta=E[(X(t_{1}))^{v_{1}}...(X(t_{k}))^{v_{k}}],\;(v_{1},...,v_{k}\in\mathbb{N}),$

we have: $f^{l}(x_{1},...,x_{k})=(x(t_{1}))^{v_{1}}...(x(t_{k}))^{v_{k}}$,

$Y_{f}(t)=(X(t_{1}+t))^{v_{1}}...(X(t_{k}+t))^{v_{k}}.$

Application of Corollary 1 to the fields $Y_{f}^{l}(t)$ brings us to the
following statement.

######  Corollary 2.

Let $X$ be a homogeneous scalar random field. Under the conditions of
Corollary 1 (with $Y^{l}_{f},M^{l}_{f,n},V^{l}_{f,n},\theta^{l}$ instead of
$X^{l},M^{l}_{n},V^{l}_{n},\mu^{l}$,respectively), for
$l=1,...,d,\;\;u=1,...,w,$

$\frac{\sum_{i=1}^{k_{n}}(f^{l}(X(t_{1}+\tau^{l,u}_{n,i}),...,X(t_{k}+\tau^{l,u}_{n,i}))-\theta^{l})}{k_{n}^{\frac{1}{2}}(V^{l}_{f,n})^{\frac{1}{2}}}\overset{P_{X,\tau}}{\Longrightarrow}Z^{l,u}.$

where all random variables $Z^{l,u}$ are standard normal and independent.

If $X$ is ergodic, the ”empirical” standard deviations
$(V^{l}_{f,n})^{\frac{1}{2}}$ can be replaced by the ”theoretical” ones
$\sigma^{l}_{f}$.

Now let $\theta:=F(t_{1},...,t_{k};x^{1},...,x^{k}):=P(X(t_{1})\leq
x^{1},...,X(t_{k})\leq x^{k})$, the $k$-dimensional distribution function of
the random field $X$ with fixed ”time” values
$t_{1},...,t_{k}\in\mathbb{R}^{m}$ and fixed space values
$x^{1},...,x^{k}\in\mathbb{R}$. Let $Y(t)=Ind\\{X(t_{1}+t)\leq
x^{1}...,X(t_{k}+t)\leq x^{k}\\}$. Note that

$E_{X}[Y(0)]=F(t_{1},...,t_{k};x^{1},...,x^{l}).$

Denote:

$M_{n}:=\frac{1}{\lambda(T_{n})}\int_{T_{n}}Y(t)\lambda(dt),\;\;\;\;V_{n}:=\linebreak\frac{1}{\lambda(T_{n})}\int_{T_{n}}(Y(t)-M_{n})^{2}\lambda(dt),$

$\textbf{t}:=\\{t_{1},...,t_{k}\\},x:=(x_{1},...,x_{k})$.

######  Corollary 3.

Let $X$ be an ergodic homogeneous scalar random field. Let
$\sqrt{k_{n}}(M_{n}-F(\textbf{t};x)))\to 0$ in $P_{X}$-probability. Consider
the family of independent random variables
$\\{\tau^{u}_{n,i},n\in\mathbb{N},\;\;1\leq i\leq k_{n},1\leq u\leq w\\}$
which, for each $n,u,i,$ are distributed uniformly on $T_{n}$. Then

$\frac{\sum_{i=1}^{k_{n}}(Ind\\{X(t_{1}+\tau^{u}_{n,i})\leq
x^{1},...,X(t_{k}+\tau^{u}_{n,i})\leq
x^{k}\\}-F(\textbf{t};x))}{k_{n}^{\frac{1}{2}}V_{n}^{\frac{1}{2}}}\overset{P_{X,\tau}}{\Longrightarrow}Z^{u},$

where all random variables $Z^{u},u=1,...,w,$ are standard normal and
independent.

The sequence $V_{n}$ may be replaced by its limit
$F(\textbf{t};x)-(F(\textbf{t};x))^{2}$.

Now we present a randomized version of the CLT when the limit normal
distribution is not standard and its covariance matrix coincides with the
marginal covariance matrix of the field X. This statement is a generalization
of the classical CLT for i.i.d. random vectors (see Theorem 29.5 in [3]) and
can be readily deduced from our Theorem 2 and 3 (with $d=1$) by using the
Cram$\acute{e}$r - Wold theorem (compare with the proof of the mentioned
Theorem 29.5).

######  Theorem 5.

Let the conditions of Theorem 1 be fulfilled and, moreover, let the field $X$
be strict sense stationary and ergodic. Let the pointwise averaging sequence
$\\{T_{n}\\}$ and the randomizing sequence $(\tau_{n,i})$ be the same for all
components $X^{l}$. Denote by $\Sigma$ the (nonsingular) covariance matrix of
the vector $X(0)$ and let $V$ be a Gaussian vector with mean $0$ and
covariance matrix $\Sigma$.

Then

$\left(k_{n}^{-\frac{1}{2}}\sum_{i=1}^{k_{n}}(X^{1}(\tau_{n,i})-M_{n}^{1}),...,k_{n}^{-\frac{1}{2}}\sum_{i=1}^{k_{n}}(X^{d}(\tau_{n,i})-M_{N}^{d})\right)\overset{P_{X,\tau}}{\Longrightarrow}V.$

If the condition (18) takes place, then also

$\left(k_{n}^{-\frac{1}{2}}\sum_{i=1}^{k_{n}}(X^{1}(\tau_{n,i})-\mu^{1}),...,k_{n}^{-\frac{1}{2}}\sum_{i=1}^{k_{n}}(X^{d}(\tau_{n,i})-\mu^{d})\right)\overset{P_{X,\tau}}{\Longrightarrow}V.$

Our theorems and Examples 1 and 2 imply the following statement.

######  Corollary 4.

1\. Let $T=\mathbb{R}^{m}$. Then Theorems 1-3 hold if $\\{T_{n}^{l}\\}$ is

\- an increasing sequence of bounded convex sets, containing balls of radii
$r_{n}\to\infty$, or

\- a sequence of homothetic sets $\\{s_{n}A\\}$ considered in Example 2.

2\. The same is true if $T=\mathbb{Z}^{m}$ and $T_{n}^{l}$ are restrictions
onto $\mathbb{Z}^{m}$ of convex sets in $\mathbb{R}^{m}$ considered above.

###  4.4 Rate of convergence

We remind that $k_{n}\in\mathbb{N},\;k_{n}\uparrow\infty,\;\\{T_{n}\\}$ is a
pointwise averaging sequence of sets and

$V_{n}:=\frac{1}{\lambda(T_{n})}\int_{T_{n}}(X(t)-M_{n})^{2}\lambda(dt).$

Let $\mathbf{F}_{n}$ be the distribution function of the normalaized sum

$Z_{n}=\frac{\sum_{i=1}^{k_{n}}(X(\tau_{n,i})-\mu)}{k_{n}^{\frac{1}{2}}(V_{n})^{\frac{1}{2}}}$

and $\Phi$ be the distribution function of the standard Gaussian law. We are
interested in the estimation of

$\Delta_{n}:=\sup_{x\in\mathbb{R}}\Delta_{n}(x),$

where

$\Delta_{n}(x)=|\mathbf{F}_{n}(x)-\Phi(x)|.$

######  Theorem 6.

Suppose that $\\{X(t),\,t\in T\\},\;X(t)\in\mathbb{R},$ is a stationary
ergodic field such that for some $\delta,\;0<\delta\leq
1,\;\mathbb{E}|X(t)|^{2+\delta}<\infty.$

There exist a constant $C$ (depending only of $\delta$) such that for all
$\varepsilon>0$ and every $n$

$\Delta_{n}\leq
P\\{V_{n}<\varepsilon\\}+Ck_{n}^{-\frac{\delta}{2}}\varepsilon^{-\frac{2+\delta}{2}}\mathbb{E}|X(0)|^{2+\delta}.$
(19)

######  Remark 3.

Below, in Lemma 3, we present several sufficient conditions for estimating the
probability $P\\{V_{n}<\varepsilon\\}.$ Their proof is easily deduced from
Theorem 4.4 in [60].

######  Lemma 3.

Let $\\{T_{n}\\}$ be a sequence of Borel sets in $T$ with the following
property: each $T_{n}$ is contained in a ball $B(b_{n},Cn)$ and
$\lambda(T_{n})\geq cn^{m}\ (b_{n}\in T,0<C<\infty,0<c<C)$. Let
$\varepsilon<\sigma^{2}$. We assume that $\mu_{4}=E[X^{4}]<\infty$. Suppose
the covariance functions:

$R_{X}(t):=E[X(t)X(0)]-\mu^{2}\;\;\;{\mathrm{a}nd}\;\;\;R_{X^{2}}(t):=E[(X(t)X(0))^{2}]-\mu_{2}^{2}$

satisfy the conditions: for some $\beta>0$

$R_{X}(t)=O(|t|^{-\beta}),\;\;\;\;R_{X^{2}}(t)=O(|t|^{-\frac{\beta}{2}}),\;\;\;t\to\infty.$

Case 1. If $\beta\leq m,$ then
$P(V_{n}<\varepsilon)\leq(\sigma^{2}-\varepsilon)^{-2}O(n^{-\frac{\beta}{2}})$.

Case 2. If $\beta=m,$ then
$P(V_{n}<\varepsilon)\leq(\sigma^{2}-\varepsilon)^{-2}O(n^{-\frac{m}{2}}(\log
n)^{\frac{1}{2}}))$.

Case 3. If $\beta>m,$ then
$P(V_{n}<\varepsilon)\leq(\sigma^{2}-\varepsilon)^{-2}O(n^{-\frac{m}{2}})$.

If we take $\varepsilon=\sigma^{2}/2,$ we get from (19) the following
corollary.

######  Corollary 5.

Under the conditions of Theorem 6 and Lemma 3 we have:
In Case 1:
$\Delta_{n}=O(\max\\{n^{-\frac{\beta}{2}},k_{n}^{-\frac{\delta}{2}}\\}).$

In Case 2: $\Delta_{n}=O(\max\\{n^{-\frac{m}{2}}(\log
n)^{\frac{1}{2}},k_{n}^{-\frac{\delta}{2}}\\}).$

In Case 3:
$\Delta_{n}=O(\max\\{n^{-\frac{m}{2}},k_{n}^{-\frac{\delta}{2}}\\}).$

Now let’s move on to the proof of the theorem 6.

###### Proof.

Since for fixed path of the process $X$ the random variables
$\\{X(\tau_{n,i})\\},$
$i=1,\ldots,n,$ are independent and identically distributed, then denoting by
$\mathbf{F}_{n}^{\tau}$ the function of conditional distribution $Z_{n},$ we
can apply the estimation ([54], Ch.5, 3, Th. 6 ):

There exists an absolute constant $C$ such that

$\Delta_{n}^{\tau}(x):=|\mathbf{F}_{n}^{\tau}(x)-\Phi(x)|\leq\frac{C}{V_{n}^{(2+\delta)/2}k_{n}^{\delta/2}}E_{\tau}\\{|X(\tau_{n,1})-M_{n}|^{2+\delta}\\}.$

Therefore

$\Delta_{n}(x)=|\mathbf{F}_{n}(x)-\Phi(x)|=|E(\mathbf{F}_{n}^{\tau}(x)-\Phi(x))|\leq
E|\Delta_{n}^{\tau}(x)|:=I_{1}+I_{2},$ (20)

where

$I_{1}=\int_{\\{V_{n}<\varepsilon\\}}|\Delta_{n}^{\tau}(x)|dP,\;\;\;I_{2}=\int_{\\{V_{n}\geq\varepsilon\\}}|\Delta_{n}^{\tau}(x)|dP.$

Evidentely

$I_{1}\leq P\\{V_{n}<\varepsilon\\}.$

Consider $I_{2}.$ As

$E_{\tau}\\{|X(\tau_{n,1})-M_{n}|^{2+\delta}\\}=\frac{1}{\lambda(T_{n})}\int_{T_{n}}|X(t)-M_{n}|^{2+\delta}\lambda(dt)\leq$
$\hskip
100.0pt2^{1+\delta}\left\\{\frac{1}{\lambda(T_{n})}\int_{T_{n}}|X(t)|^{2+\delta}\lambda(dt)+|M_{n}|^{2+\delta}\right\\},$

we have

$I_{2}\leq\frac{C}{k_{n}^{\delta/2}}\varepsilon^{-\frac{2+\delta}{2}}2^{1+\delta}\left[E|X(0)|^{2+\delta}+E|M_{n}|^{2+\delta}\right].$

Since $E|M_{n}|^{2+\delta}\leq\mathbb{E}|X(0)|^{2+\delta},$ we get the
estimation

$I_{2}\leq
C\frac{2^{2+\delta}}{k_{n}^{\delta/2}}\varepsilon^{-\frac{2+\delta}{2}}E|X(0)|^{2+\delta},$

which gives finally (19). ∎

###  4.5 Invariance principle

The next natural step is to state the functional central limit theorem
(invariance principle).

Let $X(t,\omega),t\in\mathbb{R}^{m}$, be a bi-measurable random field over a
probability space $(\Omega_{X},\mathcal{F}_{X},P_{X}).$ Let $\\{T_{n}\\}$ be a
sequence of increasing convex sets in $\mathbb{R}^{m}$ containing balls with
radii $r_{n}\to\infty.$ Let, for each $n\in\mathbb{N}$,
$\tau_{n,i},\;i=1,...,k_{n},$ be random variables over a probability space
$(\Omega_{\tau},\mathcal{F}_{\tau},P\tau)$, each being uniformly distributed
on $T_{n}$ and independent of each other and of $X$. Let
$(\Omega_{X,\tau},\mathcal{F}_{X,\tau},P_{X,\tau}):=(\Omega_{X}\times\Omega_{\tau},\mathcal{F}_{X}\times\mathcal{F}_{\tau},P_{X}\times
P_{\tau}).$

Using random variables $X(\tau_{n,i})$ , we construct in usual way the
continuous piecewise lineal random process $Z_{n}=\\{Z_{n}(t),\;t\in[0,1]\\}.$
The process $Z_{n}$ has the vertices at the points

$\left(\frac{j}{k_{n}},\;\;\frac{S_{j}}{k_{n}^{1/2}V_{n}^{1/2}}\right),\;\;\;j=0,1,\ldots,k_{n},$
(21)

where

$S_{j}=\sum_{r=1}^{j}(X(\tau_{n,r})-M_{n}),\;\;\;V_{n}=Var_{\tau}(X(\tau_{n,r})).$

Denote by ${\cal P}_{n}$ the distribution of the process $Z_{n}$ in the space
$\mathbb{C}[0,1]$ and by $W$ the distribution of standard Wiener process.

######  Theorem 7.

Under mentioned conditions:

a) For $P_{X}$-almost all $\omega$

$Z_{n}(\omega,\cdot)\overset{P_{\tau}}{\Longrightarrow}W.$ (22)

b) The convergence

$Z_{n}\overset{P_{X,\tau}}{\Longrightarrow}W.$ (23)

also takes place.

######  Remark 4.

The importance of weak convergence in (22, 23) comes from the fact that by
continuous mapping theorem (see e.g., [3]) many corollaries emerge
immediately, namely for each $W$-a.e. continuous mapping
$f:\mathbb{C}\to\mathbb{R}$ we have the convergence $f(Z_{n})\Longrightarrow
f(W)$. Tipical examples of such functionals are
$f(x)=\sup_{t\in[0,1]}x(t),\;\;f(x)=\sup_{t\in[0,1]}|x(t)|$, f(x) =
$\int_{[0,1]}h(t)dt,$ and so on…

###### Proof.

As for triangular array $\\{X(\tau_{n,i})\\}$ the Lindeberg condition is
fulfilled, by Prokhorov’s theorem (see [55], Th.3.1.), we get (22).

The relation (23) follows from (22) since the limiting measure does not depend
of $\omega.$ ∎

Let us consider now the random broken lines $\tilde{Z}_{n}$ with a different
normalization. More exactly, let

$\tilde{Z}_{n}\left(\frac{j}{k_{n}}\right)=\frac{S_{j}}{k_{n}^{1/2}\sigma^{1/2}},$

where $\sigma=Var\\{X(0)\\}.$

######  Theorem 8.

Suppose that the field $X(t)$ is ergodic. Then:

a) For $P_{X}$-almost all $\omega$

$\tilde{Z}_{n}(\omega,\cdot)\overset{P_{\tau}}{\Longrightarrow}W.$ (24)

b) The convergence

$\tilde{Z}_{n}\overset{P_{X,\tau}}{\Longrightarrow}W.$ (25)

also takes place.

###### Proof.

The proof follows from the relation
$\tilde{Z}_{n}(t)=(\frac{V_{n}}{\sigma})^{1/2}Z_{n}(t)$ and the fact that due
to the ergodic theorem $\frac{V_{n}}{\sigma}\rightarrow 1$ a.s. ∎

With a little additional restriction, we can replace $M_{n}$ by $\mu$ which is
the mean value of our field.

Define the process $\tilde{\tilde{Z}}_{n}$ as a broken line with vertices at
the points

$\left(\frac{j}{k_{n}},\;\frac{S_{j}}{k_{n}^{1/2}\sigma^{1/2}}\right),$

where now $S_{j}=\sum_{r=1}^{j}(X(\tau_{n,r}-\mu)).$

######  Theorem 9.

Suppose that the field $X(t)$ is ergodic. Additionally suppose that a.s.222See
§A for remarks on this condition.

$k_{n}^{1/2}(M_{n}-\mu)\rightarrow 0.$

Then:

a) For $P_{X}$-almost all $\omega$

$\tilde{\tilde{Z}}_{n}(\omega,\cdot)\overset{P_{\tau}}{\Longrightarrow}W.$
(26)

b) The convergence

$\tilde{\tilde{Z}}_{n}\overset{P_{X,\tau}}{\Longrightarrow}W.$ (27)

also takes place.

###### Proof.

It is sufficient to remark that a.s.

$\sup_{t\in[0,1]}|\tilde{\tilde{Z}}_{n}(t)-\tilde{Z}_{n}(t)|\leq\sigma^{-1/2}(k_{n}^{1/2}(M_{n}-\mu))\rightarrow
0.$

∎

##  5 Empirical distributions

###  5.1 Randomized versions of the Glivenko-Cantelli theorem

In this section we generalize the Glivenko-Cantelli theorem to
multidimensional distributions of stationary random processes on $\mathbb{R}$
and of homogeneous random fields on $\mathbb{R}^{m}$ (to simplify the
presentation we assume ergodicity; this assumption can be readily dropped by
the consideration of the conditional expectations as in Lemma 4 below).
Generalized versions of the Glivenko-Cantelli theorem for marginal
distributions of stationary random sequences were proved in [58, 64]. We
consider a randomized version of such theorem for ergodic homogeneous random
fields on $\mathbb{R}^{m}$.

Let $\mathbf{X}(t,\omega)=(X_{1}(t,\omega),...,X_{l}(t,\omega))$ be an
$l$-dimensional bi-measurable random field on $\mathbb{R}^{m}$ over a
probability space $(\Omega_{X},\mathcal{F}_{X},P_{X})$.

We consider the CDF of this random field:

$F(x_{1},...,x_{s}):=P(X_{1}(0)\leq x_{1},...,X_{l}(0)\leq x_{l}).$

Let $\\{T_{n}\\}$ be a sequence of increasing convex sets in $\mathbb{R}^{m}$
containing balls with radii $r_{n}\to\infty$ and let, for each
$n\in\mathbb{N}$, $\eta_{n,i},\;i=1,...,k_{n},$ be random variables over a
probability space $(\Omega_{\eta},\mathcal{F}_{\eta},P_{\eta})$, each being
uniformly distributed on $T_{n}$ and independent of each other and of $X$.
Denote:
$(\Omega_{X,\eta},\mathcal{F}_{X,\eta},P_{X,\eta}):=(\Omega_{X}\times\Omega_{\eta},\mathcal{F}_{X}\times\mathcal{F}_{\eta},P_{X}\times
P_{\eta}).$

######  Lemma 4.

If $Y(t,\omega),\;t\in\mathbb{R}^{m}$, is an homogeneous random field over the
probability space $(\Omega_{X},\mathcal{F}_{X},P_{X})$ and for some $\delta>0$

$E_{X}[|Y(0)|^{2+\delta}]<\infty,$ (28)

then with $P_{X,\eta}$-probability 1

$\lim_{n\to\infty}\frac{1}{k_{n}}\sum_{i=1}^{k_{n}}Y(\eta_{n,i},\omega)=E_{X}[Y(0)|\mathcal{I}_{X})]$

(if the random field $X$ is ergodic, $E[Y(0)|\mathcal{I}_{X})]=E_{X}[Y(0)])$.

###### Proof.

Denote $M_{n}[Y]:=\frac{1}{\lambda(T_{n})}\int_{T_{n}}Y(t)\lambda(dt)$. Since
with $P_{X}$-probability 1 for each $n\in\mathbb{N}$ the random variables
$Y(\eta_{n,i},\omega)),\;i=1,...,k_{n}$ are independent, condition (28)
implies: $P_{X}$-a.s. with $P_{\eta}$-probability 1

$\lim_{n\to\infty}\frac{1}{k_{n}}\sum_{i=1}^{k_{n}}Y(\eta_{n,i},\omega)-M_{n}[Y]]=0,$
(29)

by virtue of Corollary 1 (with p=1) and Remark 3 in [30]. The Fubini theorem
implies that (29) is valid with $P_{X,\eta}$-probability 1. By the PET, with
$P_{X}$-probability 1

$\displaystyle\lim_{n\to\infty}M_{n}[Y]=E[Y(0)|\mathcal{I}_{Y}].$ (30)

It remains to note that

$\displaystyle\frac{1}{k_{n}}\sum_{i=1}^{k_{n}}[Y(\eta_{n,i},\omega)-E[Y(0)|\mathcal{I}_{Y})]=$
$\displaystyle\left(\frac{1}{k_{n}}\sum_{i=1}^{k_{n}}Y(\eta_{n,i},\omega)-M_{n}[Y]\right)+(M_{n}-E[Y(0)|\mathcal{I}_{Y}]).$

∎

Denote:

$\displaystyle F_{n}(x^{1},...,x^{l};\omega,\eta):=$
$\displaystyle\frac{1}{k_{n}}\sum_{i=1}^{k_{n}}Ind[X_{1}(\eta_{n,i},\omega)\leq
x^{1},...,X_{l}(\eta_{n,i},\omega)\leq x^{l}].$

To simplify the notation we denote:
$\boldsymbol{x}=(x^{1},...,x^{l})\;(\in\mathbb{R}^{l})$.

######  Theorem 10.

Let the random field $\mathbf{X}(t,\omega),\;t\in R^{m},$ be ergodic and
homogeneous. With $P_{X,\eta}$-probability 1

$\lim_{n\to\infty}\underset{x\in\mathbb{R}^{l}}{\sup}|F_{n}(\boldsymbol{x};\omega,\eta)-F(\boldsymbol{x})|=0.$
(31)

###### Proof.

Since the random field $X$ is bi-measurable, the function
$(\omega,\eta)\mapsto F_{n}(\boldsymbol{x};\omega,\eta)$ is
$\mathcal{F}_{X,\eta}$-measurable. Let $\mathbb{Q}$ be the set of rational
numbers. It is clear that the function
$(\omega,\eta)\mapsto\underset{\boldsymbol{x}\in\mathbb{Q}^{l}}{\sup}|F_{n}(\boldsymbol{x};\omega,\eta)-F(\boldsymbol{x})|$
is $\mathcal{F}_{X,\eta}$-measurable, and, since $\mathbb{Q}^{l}$ is dense in
$\mathbb{R}^{l}$,

$\underset{\boldsymbol{x}\in\mathbb{R}^{l}}{\sup}|F_{n}(\boldsymbol{x};\omega,\eta)-F(\boldsymbol{x})|=\underset{\boldsymbol{x}\in\mathbb{Q}^{l}}{\sup}|F_{n}(\boldsymbol{x};\omega,\eta)-F(\boldsymbol{x})|,$

the function
$\underset{\boldsymbol{x}\in\mathbb{R}^{l}}{\sup}|F_{n}(\boldsymbol{x};\omega,\eta)-F(\boldsymbol{x})|$
is also $\mathcal{F}_{X,\eta}$-measurable.

For each $k,\;0\leq k\leq M-1,$ we denote:

$L_{k,M}=\\{\boldsymbol{x}\in\mathbb{R}^{l}:\frac{k}{M}\leq
F(\boldsymbol{x})<\frac{k+1}{M}\\}.$

Of course, if $F$ is continuous, then each $\boldsymbol{y}\in\mathbb{R}^{l}$
has at least one preimage with respect to the function $F$ and, for each $k$,
$F(L_{k,M})=[\frac{k}{M},\frac{k+1}{M})$; otherwise, some points in $[0,1]$ do
not have preimages, and, therefore, there are $k$, such that
$[\frac{k}{M},\frac{k+1}{M})\setminus F(L_{k,M})\neq\emptyset$, and even such
that $L_{k,M}=\emptyset$. If $\frac{k}{M}$ possesses at least one preimage
with respect to $F(\boldsymbol{x})$, we denote by $\boldsymbol{x}_{k,M}$ one
of these preimages; in general, if $L_{k,M}\neq\emptyset$, consider the set
$D_{k,M}:=F(L_{k,M})$, i.e., the set of all points $\boldsymbol{y}$ in
$[\frac{k}{M},\frac{k+1}{M})$ possessing preimages.

Let $0\leq k\leq{M-1}$ and $L_{k,M}\neq\emptyset$.

Denote:

$\displaystyle G(k+1,M):=P\\{(X_{1}(0),...,X_{l}(0))\in
F^{-1}([0,\frac{k+1}{M}))\\},$ $\displaystyle
H(k,M):=1-P\\{(X_{1}(0),...,X_{l}(0))\in F^{-1}([\frac{k}{M},1])\\},$

and

$\displaystyle G_{n}(k+1,M);\omega,\eta):=$
$\displaystyle\frac{1}{k_{n}}\sum_{i=1}^{k_{n}}\text{Ind}[(X_{1}(\eta_{n,i},\omega),..,X_{l}(\eta_{n,i},\omega))\in
F^{-1}([0,\frac{k+1}{M})],$ $\displaystyle H_{n}(k,M);\omega,\eta):=$
$\displaystyle
1-\frac{1}{k_{n}}\sum_{i=1}^{k_{n}}\text{Ind}[(X_{1}(\eta_{n,i},\omega),..,X_{l}(\eta_{n,i},\omega))\in
F^{-1}([\frac{k}{M},1)]].$

Note: a) if $l=1$, then
$G(k,M)=F(\boldsymbol{x}_{k+1,{M}}-0);H(k,M)=F(\boldsymbol{x}_{k,M}+0)$;

b) if the function $F$ is continuous at the point $\boldsymbol{x}_{k+1,M}$,
then $G(k+1,M)=F(\boldsymbol{x}_{k+1,M});H(k,M)=F(\boldsymbol{x}_{k,M})$.

c) if $\boldsymbol{x}\in F(L_{k,{M}})$, then $H(k,M)\leq F(\boldsymbol{x})\leq
G(k+1,M)$;
$H_{n}(k,M);\omega,\eta)\leq F_{n}(\boldsymbol{x};\omega\eta)\leq
G_{n}(k+1,M;\omega,\eta)$;

d) $0\leq G(k+1,M)-H(k,{M})\leq\frac{1}{M}$.

By Lemma 4, for each $k$ with $P_{X,\eta}$-probability 1

$\displaystyle|H_{n}(k,M;\omega,\eta)-H(k,M)|\to 0$ (32)

and

$|G_{n}(k+1,M;\omega,\eta)-G(k+1,M)|\to 0.$ (33)

If $\boldsymbol{x}\in L_{k,M}$,

$\displaystyle F_{n}(\boldsymbol{x};\omega,\eta)-F(\boldsymbol{x})\leq
G_{n}(k+1,M;\omega)-H(k,M)=$
$\displaystyle\left(G_{n}(k+1,M;\omega,\eta)-G(k+1,M)\right)+\left(G(k+1,M)-H(k,M)\right)\leq$
$\displaystyle G_{n}(k+1,M;\omega,\eta)-G(k+1,M)+\frac{1}{M}.$

Similarly,

$\displaystyle F_{n}(\boldsymbol{x};\omega,\eta)-F(\boldsymbol{x})\geq
H_{n}(k,M;\omega,\eta)-H({k,M})-\frac{1}{M}.$

Therefore, if $0\leq k\leq{[M]}$, $L_{k,M}\neq\emptyset$ and
$\boldsymbol{x}\in L_{k,M}$,

$\displaystyle|F_{n}(\boldsymbol{x};\omega,\eta)-F(\boldsymbol{x})|\leq$
$\displaystyle\max\\{|G_{n}(k+1,M);\omega,\eta)-G(k+1,M)|,|H_{n}(k,M;\omega,\eta)-H({k,M})|\\}+\frac{1}{M},$

It is clear that $\cup_{\\{0\leq k\leq{M}-1\\}}L_{k,M}=\mathbb{R}^{l}$;
therefore,

$\displaystyle\underset{\boldsymbol{x}\in\mathbb{R}^{l}}{\sup}|F_{n}(\boldsymbol{x};\omega,\eta)-F(\boldsymbol{x})|\leq$
$\displaystyle\max\\{|G_{n}(k+1,M);\omega,\eta)-G(k+1,M)|,|H_{n}(k,M;\omega,\eta)-H({k,M})|\\}+\frac{1}{M},$

and, by relations (32) and (33), with $P_{X,\eta}$-probability 1

$\displaystyle
0\leq\limsup_{n\to\infty}\underset{\boldsymbol{x}\in\mathbb{R}^{l}}{\sup}|F_{n}(\boldsymbol{x};\omega,\eta)-F(\boldsymbol{x})|\leq$
$\displaystyle\limsup_{n\to\infty}\underset{k:L_{k,M}\neq\emptyset,0\leq
k\leq{M}-1}{\max}\underset{\boldsymbol{x}\in
L_{k,M}}{\sup}|F_{n}(\boldsymbol{x};\omega,\eta)-F(\boldsymbol{x})|\leq\frac{1}{M}.$

Since $M$ is chosen arbitrarily, with $P_{X,\eta}$-probability 1

$\lim_{n\to\infty}\underset{\boldsymbol{x}\in\mathbb{R}^{l}}{\sup}|F_{n}(\boldsymbol{x};\omega,\eta)-F(\boldsymbol{x})|=0.$

∎

Let $X(t,\omega),\,t\in\mathbb{R}^{m}$, be a random field over a probability
space
$(\Omega_{X},\mathcal{F}_{X},P_{X})$.

We fix a set of ”time” points
$\boldsymbol{t}=\\{t_{1},...,t_{l}\in\mathbb{R}^{m}\\}$ where
$l\in\mathbb{N}$, and study the multidimensional CDF

$F(\boldsymbol{t};x^{1},...,x^{l}):=P(X(t_{1})\leq x^{1},...,X(t_{l})\leq
x^{l}).$

Denote:

$\displaystyle F_{n}(\boldsymbol{t};x^{1},...,x^{l};\omega,\eta):=$
$\displaystyle\frac{1}{k_{n}}\sum_{i=1}^{k_{n}}Ind[X(t_{1}+\eta_{n,i},\omega)\leq
x^{1},...,X(t_{l}+\eta_{n,i},\omega)\leq x^{l}];$

It is the $l$-dimensional randomized empirical distribution function, based on
observations of $X$ on randomly chosen points in
$\cup_{j=1}^{l}(t_{j}+T_{n})$.

We apply the latter theorem to the ergodic homogeneous random field
$\mathbf{X}(t,\omega)=(X(t_{1}+t),...,X(t_{l}+t)$ and come to the following
statement.

######  Theorem 11.

Let the random field $X(\boldsymbol{t},\omega),\;\boldsymbol{t}\in R^{m},$ be
homogeneous and ergodic. With $P_{X,\eta}$-probability 1

$\lim_{n\to\infty}\underset{(x^{1},...,x^{l})\in\mathbb{R}^{l}}{\sup}|F_{n}(\boldsymbol{t};x^{1},...,x^{l};\omega,\eta)-F(\boldsymbol{t};x^{1},...,x^{l};\omega,\eta)|=0.$
(34)

Let $\eta^{u}=\\{\eta_{n,i}^{u}\\},u=1,...,r$ be $r$ independent arrays of
randomizing random variables.

$\displaystyle F_{n}(\boldsymbol{t};x^{1},...,x^{l};\omega,\eta^{u}):=$
$\displaystyle\frac{1}{k_{n}}\sum_{i=1}^{k_{n}}Ind[\\{(\omega,\eta):X(t_{1}+\eta_{n,i}^{u},\omega)\leq
x^{1},...,X(t_{l}+\eta_{n,i}^{u},\omega)\leq x^{l}\\}];$ $\displaystyle
F_{n}^{r}(\boldsymbol{t};x^{1},...,x^{l};\omega,\eta):=\frac{1}{r}\sum_{u=1}^{r}F_{n}(\boldsymbol{t};x^{1},...,x^{l};\omega,\eta^{u}).$

######  Corollary 6.

Let the random field $\mathbf{X}(\boldsymbol{t},\omega),\;\boldsymbol{t}\in
R^{m},$ be ergodic and homogeneous. With $P_{X,\eta}$-probability 1

$\lim_{n\to\infty}\underset{x\in\mathbb{R}^{l}}{\sup}|F_{n}^{r}(\boldsymbol{t};x^{1},...,x^{l};\omega,\eta)-F(x)|=0.$
(35)

This follows from Theorem 11 and the relation:

$\displaystyle\underset{(x^{1},...,x^{l})\in\mathbb{R}^{l}}{\sup}|F_{n}^{r}(\boldsymbol{t};x^{1},...,x^{l};\omega,\eta)-F(x)|\leq$
$\displaystyle\frac{1}{r}\sum_{u=1}^{r}\underset{(x^{1},...,x^{l})\in\mathbb{R}^{l}}{\sup}|F_{n}(\boldsymbol{t};x^{1},...,x^{l};\omega,\eta^{u})-F(x)|.$

If $r\geq 2$, each estimator
$F_{n}^{r}(\boldsymbol{t};\boldsymbol{x};\omega,\eta)$ of the CDF involves
more observations and, therefore, it is more precise than the estimator
$F_{n}(\boldsymbol{t};\boldsymbol{x};\omega,\eta)$.

###  5.2 Convergence of the distributions of empirical processes

We suppose in this section that $\\{X(t,\omega),\,t\in\mathbb{R}^{m}\\},$ is a
bi-measurable stationary random field with values in $\mathbb{R}^{l}.$
Moreover we suppose that a.s. $X(t)\in(0,1)^{l},$ as the general case can be
reduced to this by the standard transformation $y^{i}=\frac{1}{\pi}\arctan
x^{i}+\frac{1}{2},\;i=1,...,l.$

Let

$G_{n}(\boldsymbol{x}):={k_{n}^{1/2}}(F_{n}(\boldsymbol{x})-F(\boldsymbol{x})),\;\;\;\boldsymbol{x}\in[0,1]^{l},$
(36)

where $F_{n}$ is the empirical distribution function for the series of r.v.
$\\{X(\tau_{n,i}),i=1,2,\ldots,k_{n}\\},$

$F_{n}(\boldsymbol{x})=\frac{1}{k_{n}}\sum_{i=1}^{k_{n}}\mathbbm{1}_{[0,\boldsymbol{x}]}(X(\tau_{n,i})),$

where $[0,\boldsymbol{x}]=\prod_{i=1}^{l}[0,x_{i}]$ and $F$ is the
distribution function of $X(0).$ Below we suppose that $F$ is continuous.

We will use also empirical processes with different centering:

$L_{n}(\boldsymbol{x})={k_{n}^{1/2}}(F_{n}(\boldsymbol{x})-M_{n}(\boldsymbol{x})),\;\;\;\boldsymbol{x}\in[0,1]^{l},$

where
$M_{n}(\boldsymbol{x})=\frac{1}{\lambda(T_{n})}\int_{T_{n}}\mathbbm{1}_{[0,\boldsymbol{x}]}(X(s))\lambda(ds).$

Let $B([0,1]^{l})$ be the set of bounded, real-valued, and measurable
functions defined on the $l$-dimensional cube $[0,1]^{l}$. Let $C([0,1]^{l})$
be the set of all continuous functions. Furthermore, let $\mathbb{D}$ be a
subset of $B([0,1]^{l})$ such that, first, it contains $C([0,1]^{l})$ and,
second, the supremum norm of every function in $\mathbb{D}$ is determined by
the supremum of the function over a countable subset of $[0,1]^{l}$.
Furthermore, assume that there is a metric $\rho$ that makes $\mathbb{D}$ a
complete separable space whose topology is weaker than the topology of uniform
convergence, and such that each time when $\rho(x_{n},x)\rightarrow 0$ and $x$
is continuous it follows that $x_{n}$ converges to $x$ uniformly. Examples of
the space $\mathbb{D}$ with various metrics/topologies with mentioned
properties can be found in a number of works (cf., e.g., [57], [23] and
references therein). For comments on weak convergence in non-separable spaces
we refer to Section 4 of [18] .

It is clear that we can consider $G_{n},L_{n}$ as random elements of
$\mathbb{D}.$

We use the notation $\mathring{W}_{F}$ for the continuous Gaussian centered
process with correlation function
$K(\boldsymbol{x},\boldsymbol{y})=F(\boldsymbol{x}\wedge\boldsymbol{y})-F(\boldsymbol{x})F(\boldsymbol{y})),\;\boldsymbol{x},\boldsymbol{y}\in[0,1]^{l},$
where $\boldsymbol{x}\wedge\boldsymbol{y}=(x_{1}\wedge
y_{1},\ldots,x_{l}\wedge y_{l}).$

Our aim is to state the weak convergence of $G_{n}$ and $L_{n}$ to
$\mathring{W}_{F}.$

######  Theorem 12.

If the field $\\{X_{t}\\}$ is ergodic, then

$L_{n}\Longrightarrow\mathring{W}_{F}.$ (37)

If additionally

$\sqrt{k_{n}}\sup_{\boldsymbol{x}\in[0,1]^{l}}|M_{n}(\boldsymbol{x})-F(\boldsymbol{x})|\overset{P}{\to}0,$
(38)

then

$G_{n}\Longrightarrow\mathring{W}_{F}.$ (39)

######  Remark 5.

Condition (38) imposes a restriction on the rate of growth of the sequence
$k_{n}$. In the case, when the a.s. convergence is considered in this
condition, the admissible growth of the integers $k_{n}$ has been studied in
[24] for iid random sequences $X(1),X(2),...$; in [67], stationary sequences
$X$ are studied, and it is established, how the admissible rate of increase of
$k_{n}$ is specified by the rate of mixing of $X$ and by rate of growth of its
metric entropy. Condition (38) encourages to study the admissible rate of the
growth of $k_{n}$ when convergence in probability is considered (we suspect it
will be higher).

We have

$\sqrt{k_{n}}\sup_{\boldsymbol{x}\in[0,1]^{l}}|M_{n}(\boldsymbol{x})-F(\boldsymbol{x})|=\sqrt{\frac{k_{n}}{\lambda(T_{n})}}\sup_{\boldsymbol{x}\in[0,1]^{l}}|\xi_{n}(\boldsymbol{x})|,$

where

$\xi_{n}(\boldsymbol{x})=\lambda(T_{n})^{-1/2}\int_{T_{n}}[\mathbbm{1}_{[0,\boldsymbol{x}]}(X(s))-F(\boldsymbol{x})]\lambda(ds).$

We see that $\xi_{n}$ is empirical process associated with initial process
$X,$ hence, in good cases 333See for example [67] which contains a large
bibliography concerning the convergence of empirical processes associated with
strictly stationary ones. $\xi_{n}$ has a continuous limit $\xi$. Then, as the
functional $h\rightarrow\sup_{\boldsymbol{x}\in[0,1]^{l}}h(\boldsymbol{x})$ is
a.e. continuous with respect to the distribution of $\xi,$
$\sup_{x\in[0,1]^{l}}|\xi_{n}(x)|$ is bounded in probability, and the
condition $k_{n}=o(\lambda(T_{n}))$ will be sufficient for (38).

We start the proof of the Theorem 12 with the following Lemma.

Let $Y_{n,j},\;j=1,\ldots,k_{n}$ be array of i.i.d. $l$-dimensional random
vectors in each row having a common distribution function $J_{n}.$ It is
supposed that $Y_{n,j}\in[0,1]^{l}$ a.s.

Consider the empirical process

$V_{n}(\boldsymbol{x})={k_{n}^{1/2}}(H_{n}(\boldsymbol{x})-J_{n}(\boldsymbol{x})),\;\;\;\boldsymbol{x}\in[0,1]^{l},$

where
$H_{n}(\boldsymbol{x})=\frac{1}{k_{n}}\sum_{i=1}^{k_{n}}\mathbbm{1}_{[0,\boldsymbol{x}]}(Y_{n,i}).$

######  Lemma 5.

Suppose additionally that $J_{n}$ converges uniformly to some continuous
distribution function $J.$

Then

$V_{n}\Longrightarrow\mathring{W}_{J}.$ (40)

######  Remark 6.

The case when $J_{n}=J$ for all $n$ is well known, see, for example, [2], [57]
or [22]. But this result for triangular arrays we could not find in the
literature and therefore we present here detailed proof. It is based on the
approach proposed in [18].

###### Proof.

At the beginning we consider the case when all one-dimensional marginal
distributions of $Y_{n,j}$ are uniform on $[0,1].$

Following [18], we need to carry out three steps: establish the convergence of
finite-dimensional distributions, check the moment condition of Theorem 1 of
[18] and, finally, estimate the modulus of continuity of the function $J_{n}.$

Step 1. The convergence of finite-dimensional distributions follows in the
standard way from Lindeberg condition and Cramer-Wold device.

Step 2. Verification of the first condition of Th.1 from [18].

We shall find it technically more convenient to work with
$\|\boldsymbol{x}\|:=\max_{1\leq i\leq l}|x_{i}|.$

We prove that for some $C>0$ and for all
$\boldsymbol{x},\boldsymbol{y}\in[0,1]^{l}$

$E|V_{n}(\boldsymbol{x})-V_{n}(\boldsymbol{y})|^{2l+2}\leq
C\|\boldsymbol{x}-\boldsymbol{y}\|^{l+1}\;\;\text{whenever}\;\;\|\boldsymbol{x}-\boldsymbol{y}\|\geq\frac{1}{k_{n}}.$
(41)

Indeed, $V_{n}(\boldsymbol{x})-V_{n}(\boldsymbol{y})$ is a normalized sum of
i.i.d. centered random variables
$\xi_{i}:=\mathbbm{1}_{[0,\boldsymbol{x}]}(Y_{n,i})-\mathbbm{1}_{[0,\boldsymbol{y}]}(Y_{n,i})-(J_{n}(\boldsymbol{x})-J_{n}(\boldsymbol{y})).$

Since the symmetric difference of parallelepipeds
$[0,\boldsymbol{x}],[0,\boldsymbol{y}]$ can be partitioned into a disjoint
union of at most $2^{l}$ parallelepipeds of the form
$[\boldsymbol{a},\boldsymbol{b}],$ it suffices to estimate the moments for the
individual elements of this partition. Let

$\alpha_{i}=\mathbbm{1}_{[\boldsymbol{a},\boldsymbol{b}]}(Y_{n,i}),\;\;\;p=P\\{Y_{n,i}\in[\boldsymbol{a},\boldsymbol{b}]\\}.$

As every parallelepipede $[\boldsymbol{a},\boldsymbol{b}]$ from our partition
has at least one edge whose length is $|x_{i}-y_{i}|$ for some $i$, then
$p\leq|J_{n}(\boldsymbol{a})-J_{n}(\boldsymbol{b})|\leq\|\boldsymbol{x}-\boldsymbol{y}\|.$

Using inequality of Th.19, [54], ch.III, we have

$E\left|\sum_{1}^{m}(\alpha_{i}-p)\right|^{2l+2}\leq
C(mE\alpha_{1}^{2l+2}+(mE\alpha_{1}^{2})^{l+1})\leq C(mp+m^{l+1}p^{l+1}),$
(42)

since $E\alpha_{1}^{2l+2}=p(1-p)[(1-p)^{2l+1}+p^{2l+1}]\leq p,$ and
$E\alpha_{1}^{2}=p(1-p)\leq p.$

Returning to $E|V_{n}(\boldsymbol{x})-V_{n}(\boldsymbol{y})|^{2l+2},$ we get
for $m=k_{n}$ and some $C>0$ the estimation

$E|V_{n}(\boldsymbol{x})-V_{n}(\boldsymbol{y})|^{2l+2}\leq
C\left(\frac{p}{k_{n}^{l}}+p^{l+1}\right).$

The right-hand side does not exceed $C\|\boldsymbol{x}-\boldsymbol{y}\|^{l+1}$
since $p\leq\|\boldsymbol{x}-\boldsymbol{y}\|$ and
$\frac{1}{k_{n}}\leq\|\boldsymbol{x}-\boldsymbol{y}\|.$

Step 3. We need to estimate
$\sqrt{k_{n}}|J_{n}(\boldsymbol{x})-J_{n}(\boldsymbol{y})|$ when
$\boldsymbol{x},\,\boldsymbol{y}$ are two adjacent points of the lattice
$\Gamma_{n}=\\{\boldsymbol{j}/k_{n}\,|\;\boldsymbol{j}\in[0,k_{n}]^{l}\\}.$
Due to the condition that marginal distributions of $Y_{n,i}$ are uniform we
get immediately

$\sqrt{k_{n}}\sup_{\boldsymbol{x},\boldsymbol{y}\in\Gamma_{n},\;\|\boldsymbol{x}-\boldsymbol{y}\|=1/k_{n}}|J_{n}(\boldsymbol{x})-J_{n}(\boldsymbol{y})|\leq
l/\sqrt{k_{n}}\rightarrow 0$

when $n\rightarrow\infty,$ which complets the proof in this case.

Now we consider the case when limiting function $J$ is continuous and
coordinatewise strictly increasing. We modify slightly the arguments from the
second part of the proof of the Th.16.4. [3].

For simplicity we use the notation $Y_{n}$ for $Y_{n,1}$ and $Y$ for a random
vector having distribution function $J.$ Let $J_{n}^{(k)},\;J^{(k)}$ be
marginal distribution functions for respectively $Y_{n}$ and $Y$:

$J_{n}^{(k)}(x_{k})=P\\{Y_{n}^{(k)}\leq
x_{k}\\},\;\;\;J^{(k)}(x_{k})=P\\{Y^{(k)}\leq x_{k}\\},\;\;x_{k}\in[0,1].$

Let $\psi_{n},\,\psi:[0,1]^{l}\to[0,1]^{l}$ be define by

$\psi_{n}(\boldsymbol{x})=(J_{n}^{(1)}(x_{1}),\ldots,J_{n}^{(l)}(x_{l})),\;\;\psi(\boldsymbol{x})=(J^{(1)}(x_{1}),\ldots,J^{(l)}(x_{l})).$

Now let
$Z_{n,i}^{(k)}=J_{n}^{(k)}(Y_{n,i}^{(k)}),\;k=1,\ldots,l,\;\;i=1,\ldots,k_{n}.$
Then vectors $Z_{n,i}=(Z_{n,i}^{(1)},\ldots,Z_{n,i}^{(l)})$ are i.i.d. and
have $[0,1]$-uniformly distributed coordinates.

Let $Z_{n}=\psi_{n}(Y_{n}),\;\;\;Z=\psi(Y).$ If
$\boldsymbol{x}_{n}\to\boldsymbol{x},$ then
$\psi_{n}(\boldsymbol{x}_{n})\to\psi(\boldsymbol{x})$ and by Th.5.5.[3] we get
the convergence $Z_{n}\Rightarrow\psi(Y)$ wich gives the convergence of
distribution functions $F_{Z_{n}}(\boldsymbol{x})\to F_{Z}(\boldsymbol{x}),$
and this convergence is uniform as $F_{Z}$ is continuous. Hence we can apply
to the triangular array $\\{Z_{n,i}\\}$ our previous result which sais that
the empirical processes $U_{n},$

$U_{n}(\boldsymbol{x})=\sqrt{k_{n}}(A_{n}(\boldsymbol{x})-F_{Z_{n}}(\boldsymbol{x})),A_{n}(\boldsymbol{x})=\frac{1}{k_{n}}\sum_{i=1}^{k_{n}}\mathbbm{1}_{[0,\boldsymbol{x}]}(Z_{n,i}),$
(43)

converge weakly to $\mathring{W}_{F_{Z}}.$

Now define two mappings $\varphi_{n},\;\varphi$ inverse to $\psi_{n},\;\psi:$

$\varphi_{n}(\boldsymbol{y})=(...,\varphi_{n}^{(k)}(y_{k}),...),\;\;\;\varphi(\boldsymbol{y})=(...,\varphi^{(k)}(y_{k}),...),$
$\varphi_{n}^{(k)}(s)=\inf\\{t\,|\,s\leq
J_{n}^{(k)}(t)\\},\;\;\varphi^{(k)}(s)=\inf\\{t\,|\,s\leq J^{(k)}(t)\\}.$

It is clear that the vectors $(Y_{n,1},\ldots,Y_{n,k_{n}})$ and
$(\varphi_{n}(Z_{n,1}),\ldots,\varphi_{n}(Z_{n,k_{n}}))$ will have the same
distribution.

Define $f_{n},f:\mathbb{D}\to\mathbb{D}$ respectively by

$(f_{n}y)(\boldsymbol{x})=y(J_{n}(\boldsymbol{x}));\;\;(fy)(\boldsymbol{x})=y(J(\boldsymbol{x}))),\;\;y\in\mathbb{D},\;\boldsymbol{x}\in[0,1]^{l}.$

If $y_{n}$ converges to $y$ in $\mathbb{D}$ and $y\in\mathbb{C}[0,1]^{l},$
then the convergence is uniform and $f_{n}(y_{n})$ will converge to $f(y)$
uniformly, due to the uniforme convergence of $J_{n}$ to $J.$ Hence, by Th.5.5
[3],

$f_{n}(U_{n})\Longrightarrow f({\mathring{W}_{F_{Z}}}),$

which gives the result since we have the equalities in distribution
$f_{n}(U_{n})\overset{d}{=}V_{n}$ and
$f(\mathring{W}_{F_{Z}})\overset{d}{=}\mathring{W}_{J}.$

To complete the proof of the lemma, it suffices bring into consideration a
family $\\{L_{\theta}\\}$ of triangular arrays, for which distribution
functions $J_{\theta}$ are continuous, strictly increasing, and uniformly
converge to the distribution function $J_{\theta_{0}}:=J$ when
$\theta\to\theta_{0}.$ Let $V_{n,\theta}$ be the empiric process associated
with $L_{\theta}$ and $V_{n,\theta_{0}}=V_{n}.$ Then we have the following
properties:

  1. 1.

For each $\theta,\;\;\;V_{n,\theta}\Rightarrow\mathring{w}_{J_{\theta}},$ as
$n\to\infty$ (by previous consideration.)

  2. 2.

For each $n,\;\;V_{n,\theta}\Rightarrow V_{n,\theta_{0},}$ as
$\;\theta\to\theta_{0}$ (due to the convergence
$J_{\theta}\to J_{\theta_{0}}.$)

  3. 3.

$\mathring{W}_{J_{\theta}}\Rightarrow\mathring{W}_{J_{\theta_{0}}},$ as
$\;\;\theta\to\theta_{0}$ (due to the convergence of covariance functions.)

From these three properties it follows immediately the convergence
$V_{n}=V_{n,\theta_{0}}\Rightarrow\mathring{W}_{J_{\theta_{0}}}=\mathring{W}_{J}$
which finishes the proof. ∎

Proof of Th.12. Remark that by Glivenko-Cantelli theorem
$M_{n}(\boldsymbol{x})=\frac{1}{\lambda(T_{n})}\int_{T_{n}}\mathbbm{1}_{[0,\boldsymbol{x}]}(X(s))\lambda(ds).$
a. s. converges uniformly to $F.$ Therefore, applying previous Lemma 5 to the
vectors $X(\tau_{n,i})$ which are i.i.d. conditionally given field $X,$ we get
the convergence :

$L_{n}\overset{P_{\tau}}{\Longrightarrow}\mathring{W}_{F},$

from which the convergence (37) follows in the usual way.

The convergence (39) follows immediately from (37) due to the condition (38):

$\sup_{\boldsymbol{x}\in[0,1]^{l}}|L_{n}(\boldsymbol{x})-G_{n}(\boldsymbol{x})|=\sqrt{k_{n}}\sup_{x\in[0,1]^{l}}|M_{n}(\boldsymbol{x})-F(\boldsymbol{x})|\overset{P}{\to}0.$

$\square$

Let $l=1.$ Applying once more the continuous mapping theorem [3], we deduce
from (37) an analog of the famous Kolmogorov’s result.

######  Corollary 7.

Let $l=1,$ $X$ be an ergodic homogeneous random field and condition (38) be
fulfilled. Then

$P\\{\sup_{y}|{k_{n}^{1/2}}(F_{n}(y)-F(y))|\leq
x\\}\rightarrow\sum_{k=-\infty}^{\infty}(-1)^{k}e^{-2k^{2}x^{2}},\;\;x>0.$

######  Remark 7.

Theorem 12 allows one to analyze the asymptotic behavior of empirical
processes associated with finite-dimensional distributions of the original
field.

Let $X(t,\omega),\,t\in\mathbb{R}^{m},\;X(t)\in\mathbb{R}^{l},$ be a random
field over a probability space $(\Omega_{X},\mathcal{F}_{X},P_{X}).$ We fixe
the points $t_{1},...,t_{l}$ and consider the distribution function

$F(\boldsymbol{x}):=P\\{X(t_{1})\leq x^{1},...,X(t_{l})\leq
x^{l}\\},\;\;\;\boldsymbol{x}=(x^{1},...,x^{l}),$

of the vector $(X(t_{1}),...,X(t_{l}))$ and associated with it the
$l$-dimensional randomized empirical distribution function $F_{n}$,

$\displaystyle
F_{n}(\boldsymbol{x}):=\frac{1}{k_{n}}\sum_{i=1}^{k_{n}}Ind[X(t_{1}+\tau_{n,i})\leq
x^{1},...,X(t_{l}+\tau_{n,i})\leq x^{l}];$

here, as before, $\\{\tau_{n,i}\\}$ is a randomizing triangular array. Let

$M_{n}(\boldsymbol{x})=E_{\tau}(F_{n}(\boldsymbol{x}))=\frac{1}{\lambda(T_{n})}\int_{T_{n}}\mathbbm{1}_{[0,\boldsymbol{x}]}(X(t_{1}+s)),...,X(t_{l}+s))\lambda(ds).$

Consider the empirical processes:

$L_{n}(\boldsymbol{x})={k_{n}^{1/2}}(F_{n}(\boldsymbol{x})-M_{n}(\boldsymbol{x})),\;\;\;\boldsymbol{x}\in[0,1]^{l},$
$G_{n}(\boldsymbol{x})={k_{n}^{1/2}}(F_{n}(\boldsymbol{x})-F(\boldsymbol{x})),\;\;\;\boldsymbol{x}\in[0,1]^{l}.$

We apply the Theorem 12 to the ergodic homogeneous random field
$\mathbf{X}(t,\omega)=(X(t_{1}+t),...,X(t_{l}+t))$ and come to the following
statement.

######  Theorem 13.

Let the random field $X(\boldsymbol{t},\omega),\;\boldsymbol{t}\in R^{m},$ be
homogeneous and ergodic. With $P_{X,\tau}$-probability 1

$L_{n}\Longrightarrow\mathring{W}_{F}.$ (44)

If additionally

$\sqrt{k_{n}}\sup_{\boldsymbol{x}\in[0,1]^{l}}|M_{n}(\boldsymbol{x})-F(\boldsymbol{x})|\overset{P}{\to}0,$
(45)

then

$G_{n}\Longrightarrow\mathring{W}_{F}.$ (46)

APPENDIX

##  Appendix A Comments on conditions (17) and (18)

In what follows we put $d=1$ for simplicity.

######  Remark 8.

The proofs of Theorems 1 and 2 suggest that, in these theorems, it is
reasonable to use sequences $\\{k_{n}\\}$ growing to $\infty$ rather fast,
i.e., to chose the ”size of randomization” as large as possible. But, in
Theorems 3 and 4, conditions (17) and (18) put some restrictions to this
growth. These conditions are related to the speed of convergence in the
Pointwise and Mean Ergodic Theorems, respectively. In the following notes we
provide an idea of these conditions. We start with well known results related
to condition (17) in the case $T=\mathbb{Z},\;\;T_{n}^{l}=\\{1,...,n\\}$.

1\. There are no universal sequences $k_{n}$ satisfying condition (17): for
each non-decreasing sequence $k_{n}\to\infty$ there is a bounded ergodic
stationary sequence $\\{X_{n}\\}$ such that $\\{k_{n}\\}$ does not satisfy
this condition [41].

2\. The sequence $k_{n}=n^{2}$ does not satisfy condition (17) [35].

3\. For each ergodic stationary random field there is a non-decreasing
sequence of integers $k_{n}\to\infty$ that satisfies condition (17) [39].

4\. Note 1 shows that the rate of convergence in the PET depends on the
properties of the field $X$; about this relation see [35, 38, 39] and
references therein.

5\. If $\sup_{n}(n^{\gamma}Var_{X}(M_{n})))<\infty$, where $\gamma>0$, then
(17) holds with $k_{n}=n^{\alpha}$, where $\alpha<\frac{\gamma}{2}$ ([14],
Proposition 1). See Notes 6 and 9 - 11 below for examples.

Now we turn to condition (18).

6\. By the Chebyshev inequality, condition (18) holds if
$k_{n}Var_{X}(M_{n})\to 0$. If the field $X$ is wide-sense ergodic, then, by
the Mean Ergodic Theorem, $Var_{X}(M_{n})\to 0$ (see Remark 1); therefore, in
this case (18) is always fulfilled if $k_{n}$ grows sufficiently slow:

$k_{n}=o((Var_{X}(M_{n}))^{-1}).$ (47)

7\. If $\\{X(k)\\}$ is a mixing stationary random sequence,
$E_{X}[(X(0))^{2}]<\infty$ and $T_{n}^{l}=\\{1,...,n\\}$, then the sequence
$k_{n}=n^{2}$ does not satisfy condition (47).

8\. Let $T=\mathbb{R}^{m}$ or $\mathbb{Z}^{m},m\geq 1$. Assume that the CLT
holds in its classical form:
$(\lambda(T_{n}))^{\frac{1}{2}}\sigma^{-1}(M_{n}-\mu)\overset{D}{\to}Z$, where
$Z$ is the standard normal random variable. We have:
$k_{n}^{\frac{1}{2}}(M_{n}-\mu)=(\frac{k_{n}\sigma^{2}}{\lambda(T_{n})})^{\frac{1}{2}}(\lambda(T_{n}))^{\frac{1}{2}}\sigma^{-1}(M_{n}-\mu)$;
therefore $k_{n}^{\frac{1}{2}}(M_{n}-\mu)\overset{D}{\to}0$, if and only if
$k_{n}=o((\lambda(T_{n})))$. A rather often case when the classical CLT fails
is when $(\lambda(T_{n}))^{\frac{1}{2}}(M_{n}-\mu)\overset{D}{\Rightarrow}0$,
hence condition (18) is fulfilled with $k_{n}=O((\lambda(T_{n})))$.

9\. Let $\\{\xi_{k}\\}$ be an ergodic stationary Markov chain with the
probability state space $(\mathcal{X},\mathcal{A},m)$ ($m$ the initial
probability measure). Let $Q(x,A)$ be the transition function, and let $F$ be
the (dense) set in $L^{2}(\mathcal{X},\mathcal{A},m)$ consisting of all
functions $f:f(x)=g(x)-\int_{\mathcal{X}}g(y)Q(x,dy),g\in
L^{2}(\mathcal{X},\mathcal{A},m)$. Then, for all $f\in F$ the following
alternative holds: the stationary random sequence $X_{k}=f(\xi_{k})$ either
satisfies the CLT in the classical form or $Var[M_{n}]=O(n^{-1})$[27]; this
alternative holds also for $f$ in a larger class of functions, which contains
$F$ (see [49]). According to the previous note, in both cases each sequence
$k_{n}=o(n)$ satisfies condition (18).

10\. Let $m\geq 1$, $X(t),t\in\mathbb{Z}^{m}$ be an wide-sense stationary
random sequence on $T$, satisfying Condition (A), $E[X(0)]=\mu$. Let $R(t)$ be
its covariance function.

Condition (18) holds if

\- $k_{n}=o(n^{a})$ and $|R(t)|=O(|t|^{-a})$, $0<a<m$, or

\- $k_{n}=o((\frac{n^{m}}{\log n}))$ and $|R(t)|=O(|t|^{-m})$, or

\- $k_{n}=o(n^{m})$ and $|R(t)|=O(|t|^{-a})$, $a>m$).

11\. Let $T=\mathbb{R}^{m}$ or $\mathbb{Z}^{m},m\geq 1$, and let
$T_{n}=(0,n]^{m},n\in\mathbb{N}$; if $X$ is wide-sense stationary and
possesses a spectral density $\psi$, which is continuous at $0$, then
condition (17) holds if $k_{n}=o(n^{m})$. This follows from the relation

$n^{m}Var_{X}\left(\int_{T_{n}}X(t)\lambda(dt)\right)=(2\pi)^{m}\psi({0})+o(1)$

(in the cases $T=\mathbb{Z}$ and $T=\mathbb{R}$ these are, respectively,
Theorems 18.2.1 and 18.3.2 in [33]; if $T=\mathbb{Z}^{m},m\geq 2,$ this is
Theorem 3 in [36]; the case $T=\mathbb{R}^{m},\\\ \;m\geq 2,$ can be treated
similarly).

12\. In the case $m=1$ various restrictions on the spectrum and the
correlation function implying a given rate of convergence in the Mean Ergodic
Theorem, hence on restrictions on the choice of $k_{n}$ in (18) are discussed
in [35, 37, 38, 40].

##  Appendix B Randomization using non-uniform
distributions

Let $X(t)=(X^{1}(t),...,X^{d}(t)),t\in T,$ be a
$\mathcal{B}\times\mathcal{F}$-measurable random field with homogeneous
components ($d\in\mathbb{N}$) and $E[|X(0)|^{2}]<\infty$ for some $\delta>0$.
For each $l=1,...,d,n\in\mathbb{N},$ consider a probability Borel measure
$\\{q_{n}^{l}\\}$ on $T_{n}^{l}$, which is the distribution of the
$T^{l}_{n}$-valued i.i.d. randomizing random vectors
$\tau^{l}_{n,i},i=1,...,k_{n},$ introduced in Subsection 1.5. By the Fubini
theorem, for each measurable function $f$ on $\mathbb{R}$ such that
$E_{P}[f(X(0))]<\infty$ we have:
$\int_{T^{l}_{n}}|f(X(t))|q_{n}^{l}(dt)<\infty$ a.s., and

$E_{\tau}[f(X(\tau_{n,i}))]=\int_{T^{l}_{n}}f(X(t))q_{n}^{l}(dt),i=1,...,k_{n}.$

In particular,

$\displaystyle
M_{n}^{l}:=E_{\tau}[X(\tau_{n,i}^{l})]=\int_{T_{n}^{l}}X(t)q_{n}^{l}(dt);$
$\displaystyle
V_{n}^{l}:=Var_{\tau}[X(\tau_{n,i}^{l})]=\int_{T_{n}^{l}}(X(t)-M_{n}^{l})^{2}q_{n}^{l}(dt).$

Hence

$\displaystyle
E_{\tau}[\sum_{i=1}^{k_{n}}f(X(\tau_{n,i}))]=k_{n}\int_{T_{n}^{l}}f(X(t))q_{n}^{l}(dt).$

######  Definition 2.

We say that the sequence of probability Borel measures $\\{q_{n}\\}$ is
_pointwise averaging_ if for each homogeneous random field $X$ the following
PET with the ”weights” $q_{n}$ holds: if $E_{X}[|f(X(0))|]<\infty$ then
$\lim_{n\to\infty}\int_{T}f(X(t))q_{n}(dt)=E_{X}[f(X(0))|\mathcal{I}_{X}]$
with probability 1; in the case, when the measures $q_{n}$ possess densities
$\varphi_{n}$ with respect to $\lambda$, we say that the sequence
$\\{\varphi_{n}\\}$ is _pointwise averaging_.

We provide a simple lemma that helps to construct pointwise averaging
sequences in $\mathbb{R}^{m}$.

######  Lemma 6.

Let $\\{\varphi_{n}\\}$ be a sequence of densities on $\mathbb{R}^{m}$ with
compact supports $S_{n}$ in $T$; if there are positive numbers $a_{n}$ such
that the sets $T_{n}^{l}:=\\{(x,y):x\in S_{n},0\leq y\leq
a_{n}\varphi_{n}(x)\\}$ form a pointwise averaging sequence in
$\mathbb{R}^{m+1}$, then $\\{\varphi_{n}\\}$ is pointwise averaging.

###### Proof.

Apply Definition 1 to the random field $Y(t,s)\equiv
X(t),t\in\mathbb{R}^{m},s\in\mathbb{R},$ (note that
$\lambda(T_{n}^{l})=a_{n}$, hence
$\frac{1}{\lambda(T_{n}^{l})}\int_{T_{n}^{l}}Y(t,s)\lambda(dtds)=\int_{S_{n}}X(t)\varphi_{n}(t)dt$;
here $\lambda$ is the Lebesgue measure on $\mathbb{R}^{m+1})$. ∎

The following two examples are implied by Lemma 6 and Examples 1 and 2 (see
also Corollaries 4.2 and 4.3 in Ch. 6 in [64]).

######  Example 3.

Let $\\{\varphi_{n}\\}$ be a sequence of densities on $\mathbb{R}^{m}$ which
are concave on their supports $S_{n}$ (the sets $S_{n}$ are compact and
convex). If the sequence $\\{S_{n}\\}$ is increasing and the sets $S_{n}$
contain balls of radii $r_{n}\to\infty$, then the sequence $\\{\varphi_{n}\\}$
is pointwise averaging; see §5.3 in [61].

######  Example 4.

Let $\\{c_{n}\\}$ be a sequence of positive numbers tending to $\infty$. If
$\varphi$ is a bounded density on $\mathbb{R}^{m}$ with a compact support $S$
containing $0$, then the sequence of _rescaled_ densities
$\varphi_{n}(x)=c_{n}^{-m}\varphi(c_{n}^{-1}x)$ is pointwise averaging.

######  Remark 9.

When $\varphi(t)>0$ for all $t\in\mathbb{R}^{m}$, some more conditions on
$\varphi$ are needed, but the class of ”good” rescaled densities is still
rather wide (see Proposition 5.3 in [61]); for example, if $\varphi$ is the
density of a nondegenerate symmetric normal distribution, then the sequence of
rescaled densities is pointwise averaging.

For more examples, related to averages by convolutions of probability
measures, see the next section.

It is easy to see that _Theorems 1 \- 5 hold for all pointwise averaging
sequences_ $\\{q_{n}^{l}\\}$ (of course, in the proofs
$\int_{T_{n}^{l}}f(X(t))q_{n}^{l}(dt)$ has to be considered instead of
$\frac{1}{\lambda(T_{n}^{l})}\int_{T_{n}^{l}}f(X(t))\lambda(dt)$).

##  Appendix C Randomized CLT on general groups

In the above study the simplest groups $\mathbb{R}^{m}$ and $\mathbb{Z}^{m}$
were considered. But the results are valid for all groups $T$ on which the PET
holds with some sequence of sets $\\{T_{n}^{l}\\}$ or ”weights”
$\\{q_{n}^{l}\\}$. Then the proof of the generalized versions of Theorems 2
and 29 may be repeated almost word for word. There is a rather rich literature
related to PETs on groups; see [44], [52], [64], [61] and the bibliographical
survey therein; the PET with exponential rates of convergence on semisimple
Lie groups is presented in [48]. In locally compact topological groups it is
natural to consider pointwise averaging sequences of sets or densities with
respect to the Haar measure. If the group is not locally compact, there is no
Haar measure on it, and one has to consider pointwise averaging with sequences
of probability Borel measures $\\{q_{n}\\}$ as ”weights” (see §B). If $T$ is a
second countable topological group and the smallest closed group containing
the support of a probability Borel measure $p$ is $T$, then the sequence
$\\{q_{n}:=\frac{1}{n}\sum_{k=1}^{n}p^{*k}\\}$ is pointwise averaging on $T$
(see Corollary 5.3 in Ch. 3, Proposition 1.1 in Ch. 5 and Theorem 6.1 in
Chapter 6 in [64]). _In all these cases the analogs of our CLTs for
homogeneous random fields with $E[(X(0)^{2}]<\infty$ are valid._

If, in addition, $p$ is symmetric (i.e. $p(A^{-1})=p(A)$ for each Borel set
$A$ in $T$) we may put $q_{n}:=p^{*n}$ (see Note 6.4 in Chapter 6 in [64] and
the references therein); the PET with these weights is valid for a homogeneous
field $X$, if $E_{X}[|X(0)|^{1+\delta}]<\infty\ \text{for some}\ \delta>0$;
note that condition

$E_{X}[|X(0)|^{2+\delta}]<\infty\ \text{for some}\ \delta>0$

guarantees that the PET holds also for the random field $(X(t))^{2}$; this, in
its turn, implies the analog of Lemma 1 and, hence, the analogs of the rest
results. Therefore, under this condition, the CLTs hold with the ”weights”
$p^{*n}$, too.

## References

  * [1] J. Beran, Y. Feng, S. Ghosh, R. Kulik, Long-memory processes, Probabilistic properties and statistical methods. Springer, Heidelberg (2013).
  * [2] P.J. Bickel, M.J. Wichura, Convergence criteria for multiparameter stochastic processes and some applications, The Ann. of Math. Stat.,42, 5 (1971) 1656–1670.
  * [3] P. Billingsley, Probability and measure. Wiley, New York (1986)
  * [4] E. Bolthausen, On the central limit theorem for stationary mixing random fields. Ann. Probab. 10 (1982), 1047–1050.
  * [5] R.C. Bradley, A remark on the central limit question for dependent random variables; J. Appl. Probability 17 (1980), 94-101.
  * [6] R. C. Bradley, Information regularity and the central limit question. Rocky Mountain J. Math. 13 (1983), 77–97.
  * [7] R. C. Bradley, Basic properties of strong mixing conditions, A survey and some open questions, Update of, and a supplement to, the 1986 original. Probab. Surv. 2 (2005), 107–144.
  * [8] R.C. Bradley, Introduction to strong mixing conditions, Vol 1-3. Kendrick Press, Huber sity, UT (2007).
  * [9] F. Browder, On the iterations of transformations in noncompact minimal dynamical systems. Proc. Amer Math. Soc. 9, 773-780 (1958).
  * [10] A. V. Bulinskiĭ, A statistical version of the central limit theorem for vector-valued random fields. (Russian) Mat. Zametki 76 (2004), no. 4, 490–501; English translation: Math. Notes 76, 455–464 (2004).
  * [11] A. V. Bulinskiĭ, I. G. Žurbenko, The central limit theorem for random fields. Dokl. Akad. Nauk SSSR 226, 23–25 (1976) (Russian); English translation: Soviet Math. Dokl. 17 (1976), 14–17.
  * [12] R. Burton, M. Denker, On the central limit theorem for dynamical systems. Trans. Amer. Math. Soc. 302 (1987), 715–-726.
  * [13] K.L. Chung, Markov chains with stationary transition probabilities, 2nd ed. Springer, Berlin (1967)
  * [14] G. Cohen, M. Lin, Laws of large numbers with rates and the one-sided ergodic Hilbert transform. Illinois J. Math. 47 (2003), 997–1031.
  * [15] Ch. Cuny, J. Dedecker, D. Volný. A functional CLT for fields of commuting transformations via martingale approximation. Zap. Nauchn. Sem. S.-Peterburg. 441, 239–262 (2015); reprinted in J. Math. Sci. (N.Y.) 219, 765–781 (2016).
  * [16] Yu. A. Davydov, On strong mixing property for Markov chains with a countable number of states, Soviet Math. Dokl. 10 (1969), 825–827.
  * [17] Yu. A. Davydov, Mixing conditions for Markov chains, Theory Prob. Appl. 12 (1973), 312–328.
  * [18] Yu. Davydov, R. Zitikis, On weak convergence of random fields, AISM (2008) 60, 345–-365.
  * [19] J. Dedecker, A central limit theorem for stationary random fields, Probab. Theory Relat. Fields 110 (1998), 397–426.
  * [20] Y. Derriennic, Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the ”central limit theorem”, Discrete Contin. Dyn. Syst. 15 (2006), 143—158.
  * [21] Y. Derriennic, M. Lin, Variance bounding Markov chains, $L_{2}$-uniform mean ergodicity and the CLT. Stochastics and Dynamics, 11 (2011), 81–94
  * [22] J. L. Doob, Stochastic processes. Reprint of the 1953 original. John Wiley & Sons, N.Y (1990).
  * [23] R. M. Dudley, Uniform Central Limit Theorems. Cambridge University Press, Cambridge (1999).
  * [24] N. Dunford, J.T. Schwartz, Linear operators, Part I, General theory. Wiley, New York (1988).
  * [25] P. Gaenssler, W. Stute, Empirical processes: a survey of results for independent and identically distributed random variables, Ann. Probability 7 (1979), 193-243
  * [26] L. Giraitis , H.L. Koul, D. Surgailis, Large sample inference for long memory processes, Imperial College Press, London (2012).
  * [27] M.I. Gordin, B.A. Lif$\check{s}$ic, Central limit theorem for stationary Markov processes, Dokl. Akad. Nauk SSSR 239 (1978), 766–767 (Russian). English translation: Soviet Math. Dokl. 19 (1978), 392–394.
  * [28] N. Herrndorf, Stationary strongly mixing sequences not satisfying the central limit theorem, Ann. of Prob. 11 (1983), 808–813.
  * [29] O. H$\ddot{a}$ggstr$\ddot{o}$m, On the central limit theorem for geometrically ergodic Markov chains. Probab. Theory Relat. Fields 132 (2005) 74–82.
  * [30] T.-C. Hu, F. Móricz, R. L. Taylor, Strong laws of large numbers for arrays of rowwise independent random variables. Acta Math. Hung. 54 (1–2) (1989), 153–162.
  * [31] I. A. Ibragimov, Some limit theorems for stationary processes. (Russian) Teor. Verojatnost. i Primenen. 7 (1962) 361–392. English translation: Theory of Probab. and Appl. 7 (1962), 349 - 382.
  * [32] I. A. Ibragimov, A remark on the central limit theorem for dependent random variables. Teor. Verojatnost. i Primenen. 20 (1975) 134–140 (Russian). English translation: Theory of Probab. and Appl. 20 (1975), 135 - 141.
  * [33] I.A. Ibragimov, Yu.V. Linnik, Independent and stationary sequences of random variables. Wolters-Noordhoff Publishing, Groningen (1971).
  * [34] A. V. Ivanov, N. N. Leonenko, Statistical analysis of random fields. Mathematics and its Applications (Soviet Series), 28. Kluwer Academic Publishers Group, Dordrecht (1989).
  * [35] A.G. Kachurovskii, The rate of convergence in ergodic theorems, Uspekhi Mat. Nauk 51 (1996) 73-124 (Russian). English translation: Russian Math. Surveys, 51 (1996) 653-703.
  * [36] A.G. Kachurovskii, Convergence of averages in the ergodic theorem for groups $\mathbb{Z}^{d}$. Zapiski Nauchnykh Seminarov POMI 256 (1999), 121–128 (Russian). English translation: J. Math. Sci. 107 (2001) 4231–4236.
  * [37] A.G. Kachurovskii, The Fejer integrals and the von Neumann ergodic theorem with continuous time, Zap. Nauchn. Sem. POMI 474 (2018) 171–182 (Russian). English translation: J. Math. Sci. 251 (2020), 111-118.
  * [38] A.G. Kachurovskii, I.V. Podvigin, Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems. Trudy Moskov. Mat. Obs. 77 (2016), 1-66 (Russian). English translation: Trans. Moscow Math. Soc 1-53 (2016) .
  * [39] A.G. Kachurovskii, I.V. Podvigin, Measuring the rate of convergence in the Birkhoff ergodic theorem. Matematicheskie zametki 106 (2019) 40–52 (Russian); English translation: Math. Notes 106 (2019) 52–62.
  * [40] A.G. Kachurovskii, A.V. Reshetenko, On the rate of convergence in von Neumann’s ergodic theorem with continuous time. Mat. Sb. 201 (2010) 25–32 (Russian); English translation in Sb. Math 201 493–500 (2010) .
  * [41] U. Krengel, Ergodic theorems. De Gruyter, Berlin (1985).
  * [42] E.L. Lemann, Elements of large-sample theory. Springer, New York (1999).
  * [43] N.N. Leonenko, On the central limit theorem for certain classes of random fields. Teor.Verojatnost. i Mat. Statis., 15 (1976) 114–122.
  * [44] E. Lindenstrauss, Pointwise theorems for amenable groups. Invent. Math. 146 (2001), 259-295.
  * [45] M. Loève, Probability theory. Van Nostrand, Princeton, N.J.-Toronto, Ont.-London (1963)
  * [46] M. El Machkouri, Kahane-Khintchine inequalities and functional central limit theorem for stationary random fields. Stochastic Process. Appl. 102, 285–299 (2002).
  * [47] M. El Machkouri, D. Volný, W.B. Wu, A central limit theorem for stationary random fields. Stochastic Process. Appl. 123 (2013) 1–14.
  * [48] G.A. Margulis, A. Nevo, E.M. Stein, Analogs of Wiener’s ergodic theorems for semisimple Lie groups. II. Duke Math. J. 103 (2000), 233–259.
  * [49] M. Maxwell, M.B. Woodroofe, Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 (2000), 713–724.
  * [50] F. Merlev$\grave{e}$de, M. Peligrad, S. Utev, Recent advances in invariance principles for stationary sequences. Probab. Surv. 3 (2006), 1–36.
  * [51] E. Mielkaitis, V. Paulauskas, Rates of convergence in the CLT for linear random fields. Lith. Math. J. 51 (2011), 233–250.
  * [52] A. Nevo, Pointwise ergodic theorems for actions of groups. Handbook of dynamical systems 1B (2006) 871–982.
  * [53] M. Peligrad, N. Zhang, On the normal approximation for random fields via martingale methods. Stochastic Process. Appl. 128 (2018), 1333–1346
  * [54] V. V. Petrov, Sums of Independent Random Variables. Springer-Verlag, Berlin and Heidelberg, (2011).
  * [55] Yu. V. Prokhorov, Convergence of Random Processes and Limit Theorems in Probability Theory, Teor. Veroyatnost. i Primenen., (1956), Volume 1, Issue 2, Pages 177–238.
  * [56] G. Samorodnitsky, Stochastic processes and long range dependence, Springer, Cham ( 2016).
  * [57] M.L. Straf, Weak convergence of stochastic processes with several parameters, Proc. 6th Berkeley Symp.,v.2 (1972) 187–221.
  * [58] W. Stute and G. Schuman, A General Glivenko-Cantelli Theorem for Stationary Sequences of Random Observations. Scand. J. Statist. 7 (1980), 102-104
  * [59] A. Tempelman, Ergodic theorems for group actions Kluwer, Dordrecht (1992).
  * [60] Tempelman, A. Randomized consistent statistical inference for random processes and fields. Statistical Inference for Stochastic Processes. v. 143, 89-105 (2022)
  * [61] A.Tempelman, A. Shulman, Dominated and Pointwise Ergodic Theorems with ”weighted” averages for bounded Lamperti Representations of Amenable Groups. J. Math.Anal.Appl. 474 (2019), 23–58.
  * [62] A. Tempelman, Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications, 143 (2022), 89-105.
  * [63] C. Tone, A central limit theorem for multivariate strongly mixing random fields. Probab. Math. Statist. 30 (2010) 215–222.
  * [64] H. G.Tucker, A generalization of the Glivenko-Cantelli theorem. Ann. Math. Statist. 30 (1959), 828–830.
  * [65] D. Volný, Y. Wang, An invariance principle for stationary random fields under Hannan’s condition. — Stochastic Process. Appl. 124, (2014), 4012–4029.
  * [66] M. I. Yadrenko, Spectral theory of random fields. Translated from the Russian. Translation Series in Mathematics and Engineering. Optimization Software, Inc., Publications Division, New York (1983).
  * [67] B. Yu, Rates of Convergence for Empirical Processes of Stationary Mixing Sequences. Ann. Probab. 22(1), (1994), 94-116.
  * [68] N. Zhang, L. Reding, M. Peligrad, On the quenched central limit theorem for stationary random fields under projective criteria. J. Theoret. Probab. 33(2020) 2351–2379.