text:
\section{Introduction and the quenched CLT}

A very interesting type of convergence, with many practical applications, is
the almost sure conditional central limit theorem and its functional form.
This means that these theorems hold when the process is not started from its
equilibrium but it is rather started from a fixed past trajectory. In the
Markovian setting such a behavior is called a limit theorem started at a
point. In general these results are known under the name of quenched limit
theorems, as opposed to the annealed ones. A quenched CLT, for instance, is a
stronger form of convergence in distribution and implies the usual CLT. There
are examples in the literature showing that the annealed CLT\ does not
necessarily implies the quenched one. See for instance Ouchti and Voln\'{y}
(2008) and Voln\'{y} and Woodroofe (2010).

The limit theorems started at a point or from a fixed past trajectory are
often encountered in evolutions in random media and they are of considerable
importance in statistical mechanics. They are also useful for analyzing Markov
chain Monte Carlo algorithms.

In the context of random processes, this remarkable property is known for a
martingale which is stationary and ergodic, as shown in Ch. 4 in Borodin and
Ibragimov (1994) or on page 520 in Derriennic and Lin (2001). By using
martingale approximations, this result was extended to larger classes of
random variables by Cuny and Peligrad (2012), Voln\'{y} and Woodroofe (2014),
Cuny and Merlev\`{e}de (2014), among others (for a survey see Peligrad, 2015).

\quad A random field consists of multi-indexed random variables $(X_{u})_{u\in
Z^{d}}$. An important class of random fields are orthomartingales which have
been introduced by Cairoli (1969) and further developed in Khoshnevisan
(1982). They have resurfaced in many recent works. New versions of the central
limit theorem for stationary orthomartingales can be found in Wang and
Woodroofe (2013), Voln\'{y} (2015, 2019), which complement the results in Basu
and Dorea (1979), where a different definition of multiparameter martingale
was used.

In order to exploit the richness of the martingale techniques several authors
provided interesting sufficient conditions for orthomartingale approximations,
such as Gordin (2009), Voln\'{y} and Wang (2014), Cuny et al. (2015), El
Machkouri and Giraudo (2016), Peligrad and Zhang (2018 a), Giraudo (2018),
Voln\'{y} (2018). Other recent results involve random fields which are
functions of independent random variables as in El Machkouri et al. (2013) and
Wang and Woodroofe (2013). Peligrad and Zhang (2018 b) obtained necessary and
sufficient conditions for an orthomartingale approximation in the mean square.
These approximations make possible to obtain the central limit theorem
(CLT)\ for a large class of random fields. As in the case of a stochastic
processes, a natural and important question is to get a quenched version of
these CLT's. Motivated by this question, we obtain first a quenched CLT for
orthomartingales. We show by examples that the situation is different for
random fields. An orthomartingale which satisfies the CLT may fail to satisfy
the quenched CLT. The example we constructed also throws light on the
optimality of the moment conditions we use in our main result. Finally, we
extend the quenched CLT to its functional form and to a larger class of random
fields which can be decomposed into a orthomartingale and a coboundary. We
shall apply our results to linear and nonlinear random fields, often
encounters in economics.

For the sake of clarity, due to the complicated notation, we shall explain in
detail the case $d=2$ and the proof of the quenched CLT. Then, in the
subsequent sections, we shall discuss the general index set $Z^{d}$ and other
extensions of these results.

Let $(\Omega,\mathcal{K},P)$ be a probability space, let $T$ and $S$ be two
commuting, invertible, bimeasurable, measure preserving transformations from
$\Omega$ to $\Omega,$ and let $\mathcal{F}_{0,0}$ be a sub-sigma field of
$\mathcal{K}$. For all $(i,j)\in Z^{2}$ define
\begin{equation}
\mathcal{F}_{i,j}=T^{-i}S^{-j}(\mathcal{F}_{0,0})\text{, }i,j\in Z.
\label{def F
\end{equation}
Assume the filtration is increasing in $i$ for every $j$ fixed and increasing
in $j$ for every $i$ fixed (i.e. $\mathcal{F}_{0,0}\mathcal{\subset F}_{0,1}$
and $\mathcal{F}_{0,0}\mathcal{\subset F}_{1,0})$. For all $i$ and $j$ we also
define the following sigma algebras generated by the unions of sigma algebras:
$\mathcal{F}_{i,\infty}=\mathcal{\vee}_{m\in Z}\mathcal{F}_{i,m},$
$\mathcal{F}_{\infty,j}=\mathcal{\vee}_{n\in Z}\mathcal{F}_{n,j}$ and
$\mathcal{F}_{\infty,\infty}=\mathcal{\vee}_{n,m\in Z}\mathcal{F}_{n,m}.$ In
addition assume the filtration is commuting, in the sense that for any
integrable variable $X,$ with notation $E_{a,b}X=E(X|\mathcal{F}_{a,b}),$ we
hav
\begin{equation}
E_{u,v}E_{a,b}X=E_{a\wedge u,b\wedge v}X.\mathbb{\ } \label{com
\end{equation}
We introduce the stationary sequence as following. Define a function
$X_{0,0}:\Omega\rightarrow R,$ which is $\mathcal{F}_{0,0}-$measurable, and
the random fiel

\begin{equation}
X_{i,j}(\omega)=X_{0,0}(T^{i}S^{j}(\omega)). \label{def X
\end{equation}
For the filtration $(\mathcal{F}_{i,j})$ defined by (\ref{def F})\ we call the
random field $(X_{i,j})_{i,j\in Z}$ defined by (\ref{def X})\ orthomartingale
difference field, if
\begin{equation}
E(X_{i,j}|\mathcal{F}_{u,v})=0\text{ if either }u<i\text{ or }v<j.
\label{ortho
\end{equation}
This definition implies that for any $i$ fixed $(X_{i,j})_{j\in Z}$ is a
sequence of martingale differences with respect to the filtration
$(\mathcal{F}_{\infty,j})_{j\in Z}$ and also for any $j$ fixed $(X_{i,j
)_{i\in Z}$ is a sequence of martingale differences with respect to the
filtration $(\mathcal{F}_{i,\infty})_{i\in Z}.$ Se
\[
S_{n,v}
{\displaystyle\sum\nolimits_{i=0}^{n-1}}
{\displaystyle\sum\nolimits_{j=0}^{v-1}}
X_{i,j}.
\]

Below, $\Rightarrow$ denotes convergence in distribution.

The results in this paper are motivated by the following annealed CLT in
Voln\'{y} (2015), which was extended to a functional CLT in Cuny et al. (2015).

\bigskip

\textbf{Theorem A }\textit{Assume that }$(X_{i,j})_{i,j\in Z}$\textit{ is
defined by (\ref{def X}) and satisfies (\ref{ortho}). Also assume that the
filtration }$(\mathcal{F}_{i,j})_{i,j\in Z}$\textit{ is defined by
(\ref{def F}) and satisfies (\ref{com}). Assume that }$S$\textit{ (or
$T$\textit{) is ergodic and }$X_{0,0}$\textit{ is square integrable,
}$E(X_{0,0}^{2})=\sigma^{2}$\textit{. Then,
\[
\frac{1}{(nv)^{1/2}}S_{n,v}\Rightarrow N(0,\sigma^{2})\text{ \textit{when}
}n\wedge v\rightarrow\infty.
\]

Let us point out that if $S$\textit{ (or }$T$\textit{) }is ergodic,
then\textit{ }the $Z^{2}$ action generated by $S$ and $T$ is necessarily
ergodic. However the ergodicity is not enough for Theorem A to hold. In
Example 5.6 in Wang and Woodroofe (2013) and then in more detail by Voln\'{y}
(2015), a simple example of ergodic random field which does not satisfy the
central limit theorem is analyzed. Starting with two sequences of i.i.d.
random variables, centered with finite second moments, $(X_{n})$ and $(Y_{n
)$, the example is provided by the random field $(Z_{i,j}),$ with
$Z_{i,j}=X_{i}Y_{j}$ for all $(i,j)$.

It should be noted that Theorem A has a different area of applications than
Theorem 1 in Basu and Dorea (1979). In this latter paper the filtration is not
supposed to be commuting. For a random field $(X_{i,j})_{i,j\geq1}$ their
filtration $(\mathcal{K}_{n,m})$ is generated by the variables $\{X_{i,j
:(j\geq1,1\leq i\leq n)\cup(i\geq1,1\leq j\leq m)\}.$ Suppose $(\xi_{i,j})$
are i.i.d., standard normal random variables. Then, Theorem A can be applied,
for instance, to the random field $(X_{i,j}),$ where $X_{i,j}(\omega
)=X_{0,0}(T^{i}S^{j}(\omega))$ with $X_{0,0}=\xi_{-1,0}\xi_{0,-1}$ and
$\mathcal{F}_{0,0}=\sigma(\xi_{i,j},i\leq0,j\leq0)$ but the result in Basu and
Dorea (1979) cannot. On the other hand the random field $(Y_{i,j}),$ defined
by $Y_{i,j}=Y_{0,0}(T^{i}S^{j}(\omega))$ with $Y_{0,0}=\sum\nolimits_{k=1
^{\infty}a_{k}(\xi_{k,0}+\xi_{0,k})$ and $\sum\nolimits_{k=1}^{\infty
|a_{k}|<\infty,$ can be treated by the result in Basu and Dorea (1979) but not
by Theorem A.

It should also be noted that Theorem A allows to study the central limit
theorem for orthomartingales which are not defined by a Bernoulli $Z^{2}$-action.

The aim of this paper is to establish a quenched version of Theorem A.

We denote by $P^{\omega}(\cdot)=P_{0,0}^{\omega}(\cdot)$ a version of the
regular conditional probability $P(\cdot|\mathcal{F}_{0,0})(\omega)$.

One of the results of this paper is the following theorem:

\begin{theorem}
\label{QCLTmart}Assume that $(X_{i,j})_{i,j\in Z}$ is defined by (\ref{def X})
and satisfies (\ref{ortho}). Also assume that the filtration $(\mathcal{F
_{i,j})_{i,j\in Z}$ is defined by (\ref{def F}) and satisfies (\ref{com}).
Assume that $S$ (or $T$) is ergodic and $X_{0,0}$ is square integrable,
$E(X_{0,0}^{2})=\sigma^{2}$. Then, for $P$-almost all $\omega\in\Omega,
\begin{equation}
\frac{1}{n}S_{n,n}\Rightarrow N(0,\sigma^{2})\text{ under }P^{\omega}.
\label{QCLTn
\end{equation}
In addition, if
\begin{equation}
E(X_{0,0}^{2}\log(1+|X_{0,0}|))<\infty, \label{moment condition
\end{equation}
then for almost all all $\omega\in\Omega,$
\begin{equation}
\frac{1}{(nv)^{1/2}}S_{n,v}\Rightarrow N(0,\sigma^{2})\text{ under }P^{\omega
}\text{ when }n\wedge v\rightarrow\infty. \label{QCLTnm
\end{equation}

\end{theorem}

We would like to mention that, because by integration the quenched CLT implies
the annealed CLT, the conclusion in Theorem \ref{QCLTmart} implies the CLT in
Theorem A. However, when the summation on the rectangles is not restricted,
the integrability assumption (\ref{moment condition})\ is stronger than in
Theorem A. Later on, in Theorem \ref{Th funct CLT}, we shall extend this
result to a functional central limit theorem. \ Let us also notice that the
second part of Theorem \ref{QCLTmart} does not always hold under the
assumption $E(X_{0,0}^{2})<\infty$. As a matter of fact we are going to
provide an example to support this claim.

\begin{theorem}
\label{Counterex}Under the setting used in Theorem \ref{QCLTmart}, there is a
stationary sequence $(X_{n,m})_{n,m\in Z}$\ satisfying (\ref{ortho}), adapted
to a commuting filtration $(\mathcal{F}_{i,j})_{i,j\in Z},$ with
$E(X_{0,0}^{2}\ln(1+|X_{0,0}|))=\infty,$ for any $0<\varepsilon<1,$
$E(X_{0,0}^{2}\ln^{1-\varepsilon}(1+|X_{0,0}|))<\infty$ and such that
$(S_{n,m}/\sqrt{nm})_{(n,m)\in Z^{2}}$ does not satisfy the quenched CLT in
(\ref{QCLTnm}).
\end{theorem}

We mention that, as a matter of fact, in our examples, both transformations
constructed for the definition of $(X_{n,m})_{n,m\in Z}$ and for the
filtration $(\mathcal{F}_{i,j})_{i,j\in Z},$ are ergodic. Also, this example
satisfies the quenched CLT in (\ref{QCLTn}).

The detailed proofs of these two theorems are contained in Section
\ref{proof main}. Various extensions of Theorem \ref{QCLTmart} will be given
in subsequent sections.

In Section \ref{functional CLT} we formulate the functional form of the
quenched CLT and we indicate how to prove it, by adapting the arguments from
the proof of Theorem \ref{QCLTmart} and some other proofs of several known results.

For the sake of applications, in Section \ref{cob}, we extend the results
beyond orthomartingales, to a class of random fields which can be decomposed
into an orthomartingale and a generalized coboundary.

In Section \ref{section gend} we show that Theorem \ref{QCLTmart} remains
valid for random fields indexed by $Z^{d},$ $d>2.$ The only difference is that
we replace condition (\ref{moment condition}) by $E(X_{0,0}^{2}\log
^{d-1}(1+|X_{0,0}|))<\infty.$

In Section \ref{Appl} we apply our results to linear and nonlinear random
fields with independent innovations. Several useful results for our proofs are
given in Section 7.

\section{Proofs of Theorems \ref{QCLTmart} and \ref{Counterex}
\label{proof main}}

\textbf{Proof of Theorem \ref{QCLTmart}}

\bigskip

To fix the ideas, let us suppose that the transformation $S$ is ergodic. Let
us denote by $\hat{T}$ and $\hat{S}$ the operators on $L_{2}$, defined by
$\hat{T}f=f\circ T$ and $\hat{S}f=f\circ S.$ Everywhere in the paper, for $x$
real, we shall denote by $[x]$ the integer part of $x.$

By using a truncation argument, we show first that, without restricting the
generality, we can prove the theorem under the additional assumption that the
variables are bounded. We shall introduce the following projection operators
\[
\mathcal{P}_{i,j}(X)=E_{i,j}(X)-E_{i,j-1}(X)-E_{i-1,j}(X)+E_{i-1,j-1}(X).
\]
Let $A$ be a positive integer. Denote $X_{i,j}^{^{\prime}}=X_{i,j
I(|X_{i,j}|\leq A)$ and $X_{i,j}^{"}=X_{i,j}I(|X_{i,j}|>A).$ Therefore, we can
represent $(X_{i,j})$ as a sum of two orthomartingale differences adapted to
the same filtration.
\begin{equation}
X_{i,j}=\mathcal{P}_{i,j}(X_{i,j}^{^{\prime}})+\mathcal{P}_{i,j}(X_{i,j}^{"}).
\label{truncation
\end{equation}
Note that
\[
|\mathcal{P}_{0,0}(X_{0,0}^{"})|\leq|X_{0,0}|+E_{-1,0}|X_{0,0}|+E_{0,-1
|X_{0,0}|+E_{-1,1}|X_{0,0}|.
\]
Whence, by the properties of conditional expectation, $E(X_{0,0})^{2}<\infty$
implies
\begin{equation}
E(\mathcal{P}_{0,0}(X_{0,0}^{"}))^{2}<\infty\label{secmp
\end{equation}
and $E(X_{0,0}^{2}\log(1+|X_{0,0}|))<\infty$ implies
\begin{equation}
E((\mathcal{P}_{0,0}(X_{0,0}^{"}))^{2}\log(1+|(\mathcal{P}_{0,0}(X_{0,0
^{"}))|)<\infty. \label{secmp2
\end{equation}
Se
\[
S_{n,v}^{^{\prime}}
{\displaystyle\sum\nolimits_{i=0}^{n-1}}
{\displaystyle\sum\nolimits_{j=0}^{v-1}}
\mathcal{P}_{i,j}(X_{i,j}^{^{\prime}})\text{ and }S_{n,v}^{"}
{\displaystyle\sum\nolimits_{i=0}^{n-1}}
{\displaystyle\sum\nolimits_{j=0}^{v-1}}
\mathcal{P}_{i,j}(X_{i,j}^{"}).
\]
We shall show that, for $P-$almost all $\omega,$
\[
\lim_{A\rightarrow\infty}\text{ }\lim\sup_{n\wedge v\rightarrow\infty
}P^{\omega}(\frac{1}{(nv)^{1/2}}|S_{n,v}^{"}|>\varepsilon)=0.
\]
By conditional Markov inequality, it is enough to show that
\begin{equation}
\lim_{A\rightarrow\infty}\lim_{n\wedge v\rightarrow\infty}\frac{1}{nv
E_{0,0}(S_{n,v}^{"})^{2}=0\text{ \ \ a.s.} \label{negli1
\end{equation}
By the orthogonality of the orthomartingale differences, we have that
\begin{equation}
\frac{1}{nv}E_{0,0}((S_{n,v}^{"})^{2})=\frac{1}{nv
{\displaystyle\sum\nolimits_{i=0}^{n-1}}
{\displaystyle\sum\nolimits_{j=0}^{v-1}}
E_{0,0}(\mathcal{P}_{i,j}(X_{i,j}^{"}))^{2}. \label{orthogonality
\end{equation}
Note that the conditional expectation introduces a family of operators defined
by
\[
Q_{1}(f)=E_{0,\infty}(\hat{T}f)\text{ ; }Q_{2}(f)=E_{\infty,0}(\hat{S}f).
\]
So, using (\ref{com}), we can write
\[
E_{0,0}(\mathcal{P}_{i,j}(X_{i,j}^{"}))^{2}=Q_{1}^{i}Q_{2}^{j}(\mathcal{P
_{0,0}(X_{0,0}^{"}))^{2}.
\]

Since $Q_{1}$ and $Q_{2}$ are integral preserving Dunford-Schwartz operators,
by the ergodic theorem (see Theorem 3.5 in Ch. 6 in Krengel, 1985), if we
assume finite second moment, by (\ref{secmp})
\[
\lim_{n\rightarrow\infty}\frac{1}{n^{2}
{\displaystyle\sum\nolimits_{i=0}^{n-1}}
{\displaystyle\sum\nolimits_{j=0}^{n-1}}
Q_{1}^{i}Q_{2}^{j}(\mathcal{P}_{0,0}(X_{0,0}^{"}))^{2}=E(\mathcal{P
_{0,0}(X_{0,0}^{"}))^{2}\text{ a.s.
\]
If we assume $E(X_{0,0}^{2}\log(1+|X_{0,0}|))<\infty$ then, by (\ref{secmp2})
and Theorem 1.1 in Ch. 6, Krengel (1985), we obtain
\begin{equation}
\lim_{n\wedge v\rightarrow\infty}\frac{1}{nv
{\displaystyle\sum\nolimits_{i=0}^{n-1}}
{\displaystyle\sum\nolimits_{j=0}^{v-1}}
Q_{1}^{i}Q_{2}^{j}(\mathcal{P}_{0,0}(X_{0,0}^{"}))^{2}=E(\mathcal{P
_{0,0}(X_{0,0}^{"}))^{2}\text{ a.s.} \label{erg
\end{equation}
Clearly $\lim_{A\rightarrow\infty}\mathcal{P}_{0,0}(X_{0,0}^{"})=0$ a.s. So,
by the dominated convergence theorem,
\[
\lim_{A\rightarrow\infty}E(\mathcal{P}_{0,0}(X_{0,0}^{"}))^{2}=0,
\]
and (\ref{negli1}) is established. By Theorem 3.2 in Billingsley (1999), in
order to establish conclusion (\ref{QCLTnm}) of Theorem \ref{QCLTmart}, it is
enough to show that for $A$ fixed, for almost all $\omega\in\Omega,
\[
\frac{1}{(nv)^{1/2}}S_{n,v}^{\prime}\Rightarrow N(0,\sigma_{A}^{2})\text{
under }P^{\omega}\text{ as }n\wedge v\rightarrow\infty,\text{ and }\sigma
_{A}^{2}\rightarrow\sigma^{2}\text{ as }A\rightarrow\infty.
\]
Above, $\sigma_{A}^{2}=E(\mathcal{P}_{0,0}(X_{0,0}^{^{\prime}}))^{2}.$
Clearly, when $A\rightarrow\infty,$ $\sigma_{A}^{2}\rightarrow\sigma^{2}.$
Therefore the result is established if we prove Theorem \ref{QCLTmart}\textbf{
}for orthomartingale differences which are additionally uniformly bounded.

So, in the rest of the proof, without restricting the generality, we shall
assume that the variables $(X_{i,j})_{i,j\in Z}$ are bounded by a positive
constant $C$. Also, proving the result for $n>v\rightarrow\infty$ is
equivalent to proving it for any subsequence $(n,v_{n})$ with $v_{n
\rightarrow\infty$ as $n\rightarrow\infty$. To ease the notation we shall
denote $v=v_{n}$.

Denote
\begin{equation}
F_{i,v}=\frac{1}{v^{1/2}
{\displaystyle\sum\nolimits_{j=0}^{v-1}}
X_{i,j}. \label{defFiv
\end{equation}
We treat the double summation as a sum of a triangular array of martingale
differences $(F_{i,v})_{i\geq0}:
\[
\frac{1}{(nv)^{1/2}}S_{n,v}=\frac{1}{n^{1/2}
{\displaystyle\sum\nolimits_{i=0}^{n-1}}
F_{i,v}.
\]
We shall apply Theorem 1 in G\"{a}nssler and H\"{a}usler (1979), given for
convenience in Theorem\ \ref{cltGH} from Section \ref{AUX}, to $D_{n,i
=F_{i,v}/\sqrt{n}$. We have to show that for almost all $\omega,$ both
conditions of this theorem are satisfied. Namely we shall verify that $P-$for
almost all $\omega\in\Omega$ and all rationals $q\in\lbrack0,1]$
\begin{equation}
\lim_{n\rightarrow\infty}\frac{1}{n}E_{0,0}|\sum\nolimits_{i=0}^{[(n-1)q]
(F_{i,v}^{2}-\sigma^{2})|=0\text{. } \label{limit cond
\end{equation}
an
\begin{equation}
\frac{1}{n}E_{0,0}\max_{0\leq i\leq n-1}F_{i,v}^{2}\text{ is bounded.}
\label{max bound cond
\end{equation}

We verify first (\ref{limit cond}). Note that, since the rationals are
countable, it is enough to show that for any $q$ rationa
\[
\lim_{n\rightarrow\infty}\frac{1}{n}E_{0,0}|\sum\nolimits_{i=0}^{[(n-1)q]
(F_{i,v}^{2}-\sigma^{2})|=0\text{ \ }P-\text{a.s.
\]
\ We verify it first with $q=1$ and use a blocking procedure.

Let $m\geq1$ be a fixed integer and define consecutive blocks of indexes of
size $m$, $I_{j}(m)=\{(j-1)m,...,mj-1\}.$ In the set of integers from $0$ to
$n-1$ we have $u=u_{n}(m)=[n/m]$ such blocks of integers and a last one
containing less than $m$ indexes. Practically, by the triangle inequality, we
writ
\begin{gather*}
\frac{1}{n}|\sum\nolimits_{i=0}^{n-1}(F_{i,v}^{2}-\sigma^{2})|\leq\\
\frac{1}{n}\sum\nolimits_{j=1}^{u}|\sum\nolimits_{k\in I_{j}(m)}(F_{k,v
^{2}-\sigma^{2})|+\frac{1}{n}|\sum\nolimits_{k=um}^{n-1}(F_{k,v}^{2
-\sigma^{2})|\leq\\
\frac{1}{u}\sum\nolimits_{j=1}^{u}|\frac{1}{m}\sum\nolimits_{k\in I_{j
(m)}F_{k,v}^{2}-\sigma^{2}|+\frac{1}{n}|\sum\nolimits_{k=um}^{n-1}(F_{k,v
^{2}-\sigma^{2})|=\\
I_{n,m}+II_{n,m}.
\end{gather*}
The task is now to show that
\begin{equation}
\lim_{m\rightarrow\infty}\lim_{n\wedge v\rightarrow\infty}E_{0,0
(I_{n,m})=0\text{ }\ \text{a.s.} \label{one
\end{equation}
an
\begin{equation}
\lim_{m\rightarrow\infty}\lim_{n\wedge v\rightarrow\infty}E_{0,0
(II_{n,m})=0\text{ }\ \text{a.s.} \label{two
\end{equation}
Let us treat first the limit of $E_{0,0}(I_{n,m})$. Let $N_{0}$ be a fixed
integer and consider $n\wedge v>N_{0}$. By using the properties of the
conditional expectations and (\ref{com}) we obtain the following bound for
$E_{0,0}(I_{n,m}):$
\begin{gather*}
E_{0,0}(I_{n,m})=\frac{1}{u}E_{0,0}\sum\nolimits_{j=1}^{u}|\frac{1}{m
\sum\nolimits_{k\in I_{j}(m)}F_{k,v}^{2}-\sigma^{2}|\\
=\frac{1}{u}E_{0,0}\sum\nolimits_{j=1}^{u}E_{(j-1)m,0}|\frac{1}{m
\sum\nolimits_{k\in I_{j}(m)}F_{k,v}^{2}-\sigma^{2}|\\
=E_{0,0}\frac{1}{u}\sum\nolimits_{i=0}^{u-1}\hat{T}^{im}E_{0,0}|\frac{1
{m}\sum\nolimits_{k=0}^{m-1}F_{k,v}^{2}-\sigma^{2}|\\
\leq E_{0,0}\frac{1}{u}\sum\nolimits_{i=0}^{u-1}\hat{T}^{im}(h_{m,N_{0}}),
\end{gather*}
where we have used the notatio
\[
h_{m,N_{0}}=\sup_{v>N_{0}}E_{0,0}|\frac{1}{m}\sum\nolimits_{k=0}^{m-1
F_{k,v}^{2}-\sigma^{2}|.
\]
Note that $h_{m,N_{0}}$ is bounded. Indeed, by the martingale property and the
uniform boundedness of the variables by $C$, it follows that
\begin{gather*}
h_{m,N_{0}}\leq\sigma^{2}+\frac{1}{m}\sum\nolimits_{k=0}^{m-1}\sup_{v>N_{0
}E_{0,0}(F_{k,v}^{2})\\
=\sigma^{2}+\frac{1}{m}\sum\nolimits_{k=0}^{m-1}\sup_{v>N_{0}}E_{0,0}(\frac
{1}{v}\sum\nolimits_{u=0}^{v-1}X_{k,u}^{2})\leq\sigma^{2}+C^{2}.
\end{gather*}
By the ergodic theorem, (see Theorem 11.4 in Eisner et al., 2015 or Corollary
3.8 in Ch. 3, Krengel, 1985) for each $m$ and $N_{0}$
\[
\lim_{u\rightarrow\infty}\frac{1}{u}\sum\nolimits_{i=0}^{u-1}\hat{T
^{im}h_{m,N_{0}}=E(h_{m,_{N_{0}}}|I)=E_{I}(h_{m,_{N_{0}}})\text{ a.s.,
\]
where $I$ is the invariant sigma field for the operator $T$. Furthermore, we
also have that
\[
\frac{1}{u}\sum\nolimits_{i=0}^{u-1}\hat{T}^{im}h_{m,N_{0}}\leq\sigma
^{2}+C^{2}.
\]
So, by Theorem 34.2 (v) in Billingsley (1995) (see Theorem \ref{Bill} in
Section \ref{AUX}) we derive that
\[
\lim_{u\rightarrow\infty}E_{0,0}\frac{1}{u}\sum\nolimits_{i=0}^{u-1}\hat
{T}^{im}h_{m,N_{0}}=E_{0,0}E_{I}(h_{m,_{N_{0}}})\text{ a.s.
\]
Since the functions are bounded, by applying twice, consecutively, Theorem
\ref{Bill}, we obtain tha
\[
\lim_{N_{0}\rightarrow\infty}\lim_{u\rightarrow\infty}E_{0,0}\frac{1}{u
\sum\nolimits_{i=0}^{u-1}\hat{T}^{im}h_{m,N_{0}}=E_{0,0}E_{I}(\lim
_{N_{0}\rightarrow\infty}h_{m,_{N_{0}}})\text{ a.s.
\]
Clearly, because the variables are bounded, for every $m$ fixed
\begin{align*}
E_{0,0}E_{I}(\lim_{N_{0}\rightarrow\infty}h_{m,_{N_{0}}})  &  =E_{0,0
E_{I}(\lim\sup_{v}E_{0,0}|\frac{1}{m}\sum\nolimits_{k=0}^{m-1}F_{k,v
^{2}-\sigma^{2}|)\\
&  \leq E_{0,0}E_{I}E_{0,0}(\lim\sup_{v}E_{\infty,0}|\frac{1}{m
\sum\nolimits_{k=0}^{m-1}F_{k,v}^{2}-\sigma^{2}|).
\end{align*}
Now, by using again the fact that the variables are bounded and using Theorem
\ref{Bill}, in order to show tha
\[
\lim_{m\rightarrow\infty}E_{0,0}E_{I}(\lim_{N_{0}\rightarrow\infty
}h_{m,_{N_{0}}})=0\text{ }P\text{-a.s.
\]
it is enough to show that
\begin{equation}
\lim_{m\rightarrow\infty}\lim\sup_{v}E_{\infty,0}|\frac{1}{m}\sum
\nolimits_{k=0}^{m-1}F_{k,v}^{2}-\sigma^{2}|=0\text{ }\ \text{a.s.}
\label{to show
\end{equation}
With this aim, we note first that by the ergodicity of $S$ and the fact that
the variables are bounded, it follows that, for any $k,$
\begin{equation}
\lim_{v\rightarrow\infty}E_{\infty,0}F_{k,v}^{2}=\lim_{v\rightarrow\infty
}\frac{1}{v}E_{\infty,0}(\sum\nolimits_{j=0}^{v-1}X_{k,j}^{2})=\sigma^{2}.
\label{erg2
\end{equation}
Denote $P_{\infty,0}^{\omega}(\cdot)=P(\cdot|\mathcal{F}_{\infty,0}).$ We also
know that for any $k$, by the quenched CLT for stationary martingale
differences (see, for instance, Ch. 4 in Borodin and Ibragimov (1994) or
Derrienic and Lin (2001)), for almost all $\omega,$ $F_{k,v}\Rightarrow N_{k}$
under $P_{\infty,0}^{\omega}$, where $N_{k}$ is a centered normal random
variable with variance $\sigma^{2}.$ Therefore, by the sufficiency part of the
convergence of moments associated to weak convergence, namely Theorem 3.6\ in
Billingsley (1999), we have tha
\begin{equation}
(F_{k,v}^{2})_{v\geq1}\text{ is uniformly integrable under}\ P_{\infty
,0}^{\omega}\text{ for almost all }\omega. \label{U.I
\end{equation}
By the functional quenched CLT\ for martingales (see Ch. 4 in Borodin and
Ibragimov (1994)), for almost all $\omega$,$\ $we know that
\[
(F_{0,v},F_{1,v},...,F_{m-1,v})\Rightarrow(N_{0},N_{1},...,N_{m-1})\text{
under }P_{\infty,0}^{\omega}\text{ as }v\rightarrow\infty,
\]
where $(N_{0},N_{1},...,N_{m-1})$ is a Gaussian vector of centered normal
variables with variance $\sigma^{2}$. But since $(F_{j,v})_{j\in Z}$ are
uncorrelated it follows by (\ref{U.I}) that the variables in $(N_{i})_{i\geq
0}$ are also uncorrelated and therefore $(N_{i})_{i\geq0}$ is an i.i.d.
sequence. By the continuous mapping theorem,
\[
\frac{1}{m}\sum\nolimits_{k=0}^{m-1}(F_{k,v}^{2}-\sigma^{2})\Rightarrow
\frac{1}{m}\sum\nolimits_{k=0}^{m-1}(N_{k}^{2}-\sigma^{2})\text{ under
}P_{\infty,0}^{\omega}\text{ for almost all }\omega\text{.
\]
By (\ref{U.I}) it follows that $(\sum\nolimits_{k=0}^{m-1}(F_{k,v}^{2
-\sigma^{2}))_{v\geq1}$ is also uniformly integrable, so we can apply the
convergence of moments from Theorem 3.5\ in Billingsley (1999). Therefore,
denoting by $\mathcal{E}$ the expectation in rapport with the probability on
the space where the variables $(N_{k})^{\prime}s$ are defined, we obtain
\[
\lim_{v\rightarrow\infty}E_{\infty,0}|\frac{1}{m}\sum\nolimits_{k=0
^{m-1}(F_{k,v}^{2}-\sigma^{2})|=\mathcal{E}|\frac{1}{m}\sum\nolimits_{k=0
^{m-1}(N_{k}^{2}-\sigma^{2})|\text{ \ }\ \text{a.s.
\]
By letting $m\rightarrow\infty$ and using the law of large numbers for an
i.i.d. sequence, we obtai
\[
\lim_{m\rightarrow\infty}\mathcal{E}(|\frac{1}{m}\sum\nolimits_{k=0
^{m-1}(N_{k}^{2}-\sigma^{2})|=0.
\]
Therefore (\ref{to show}) follows. As a consequence, we obtain (\ref{one}).

In order to treat the term (\ref{two}), we estimat
\begin{gather*}
E_{0,0}(II_{n,m})=E_{0,0}\frac{1}{n}|\sum\nolimits_{k=um}^{n-1}(F_{k,v
^{2}-\sigma^{2})|\leq\frac{m}{n}\sigma^{2}+E_{0,0}\frac{1}{n}\sum
\nolimits_{k=um}^{n-1}F_{k,v}^{2}\\
\leq\frac{m}{n}\sigma^{2}+\frac{1}{n}\sum\nolimits_{k=um}^{n-1}\frac{1}{v
{\displaystyle\sum\nolimits_{j=0}^{v-1}}
E_{0,0}X_{k,j}^{2}\leq\frac{m}{n}(\sigma^{2}+C^{2})\text{ a.s.
\end{gather*}
Whence, (\ref{two})\ follows, by passing to the limit first with
$n\rightarrow\infty$ followed by $m\rightarrow\infty$.

Overall, we have shown tha
\[
\lim_{n\wedge v\rightarrow\infty}\frac{1}{n}E_{0,0}|\sum\nolimits_{u=0
^{n-1}(F_{u,v}^{2}-\sigma^{2})|=0\text{ a.s.
\]
If we replace now $n-1$ by $[(n-1)q]$, with $q$ a rational number, we easily
see that we also have convergence to $q\sigma^{2}$ and (\ref{limit cond}) follows.

It remains to verify the second condition of Theorem \ref{cltGH}, namely to
prove (\ref{max bound cond}). To show it, note that, by the martingale
property,
\begin{align*}
\frac{1}{n}E_{0,0}(\max_{0\leq i\leq n-1}F_{i,v}^{2})  &  \leq\frac{1
{n}E_{0,0}
{\displaystyle\sum\nolimits_{i=0}^{n-1}}
F_{i,v}^{2})\\
&  =\frac{1}{nv}
{\displaystyle\sum\nolimits_{i=0}^{n-1}}
{\displaystyle\sum\nolimits_{u=0}^{v-1}}
E_{0,0}(X_{i,u}^{2}))\leq C^{2}\text{ a.s.
\end{align*}
The proof of the theorem is now complete. $\square$

\bigskip

\textbf{Proof of Theorem \ref{Counterex}}

\bigskip

We start with an i.i.d. random field $(\xi_{n,m})_{n,m\in Z}$ defined on a
probability space $(\Omega,\mathcal{K},P)$ with the distribution
\begin{equation}
P(\xi_{0,0}=-1)=P(\xi_{0,0}=1)=1/2. \label{defcsi
\end{equation}
Without restricting the generality we shall define $(\xi_{\mathbf{u
})_{\mathbf{u}\in Z^{2}}$ in a canonical way on the probability space $\Omega$
$=R^{Z^{2}}$, endowed with the $\sigma-$field $\mathcal{B},$ generated by
cylinders. Then, if $\omega=(x_{\mathbf{v}})_{\mathbf{v}\in Z^{2}}$, we define
$\xi_{\mathbf{u}}^{\prime}(\omega)=x_{\mathbf{u}}$. We construct a probability
measure $P^{\prime}$ on $\mathcal{B}$ such that for all $B\in\mathcal{B}$, any
$m$ and $\mathbf{u}_{1},...,\mathbf{u}_{m}$ we hav
\[
P^{\prime}((x_{\mathbf{u}_{1}},...,x_{\mathbf{u}_{m}})\in B)=P((\xi
_{\mathbf{u}_{1}},...,\xi_{\mathbf{u}_{m}})\in B).
\]
The new sequence $(\xi_{\mathbf{u}}^{\prime})_{\mathbf{u}\in Z^{2}}$ is
distributed as $(\xi_{\mathbf{u}})_{\mathbf{u}\in Z^{2}}$ and re-denoted by
$(\xi_{\mathbf{u}})_{\mathbf{u}\in Z^{2}}$. We shall also re-denote
$P^{\prime}$ as $P.$ Now on $R^{Z^{2}}$ we introduce the operator
\[
T_{\mathbf{u}}((x_{\mathbf{v}})_{\mathbf{v}\in Z^{2}})=(x_{\mathbf{v+u
})_{\mathbf{v}\in Z^{2}}.
\]
Two of them will play an important role, namely when $\mathbf{u=}(1,0)$ and
when $\mathbf{u=}(0,1).$ By interpreting the indexes as notations for the
lines and columns of a matrix, we shall cal

\[
T((x_{u,v})_{(u,v)\in Z^{2}})=(x_{u+1,v})_{(u,v)\in Z^{2}
\]
the vertical shift an
\[
S((x_{u,v})_{(u,v)\in Z^{2}})=(x_{u,v+1})_{(u,v)\in Z^{2}
\]
the horizontal shift. Introduce the filtration $\mathcal{F}_{n,m}=\sigma
(\xi_{i,j},i\leq n,j\leq m)$ and notice that this filtration is commuting. We
assume $\mathcal{K}=\mathcal{F}_{\infty,\infty}.$ The transformations $T$ and
$S$ are invertible, measure preserving, commuting and ergodic. Furthermore
$T_{i,j}=T^{i}S^{j}.$

For a measurable function $f$ defined on $R^{Z^{2}}$ define
\begin{equation}
X_{j,k}=f(T^{j}S^{k}(\mathbf{\xi}_{a,b})_{a\leq0,b\leq0}). \label{defXfield
\end{equation}
We notice that the variables are adapted to the filtration $(\mathcal{F
_{n,m})_{n,m\in Z}$.

\bigskip

As an important step for constructing our example we shall establish the
following lemma:

\begin{lemma}
\label{Rohling2}For every $n$ and every $\varepsilon>0$ we can find a set
$F=F(n,\varepsilon)$ which is $\mathcal{F}_{0,0}$ measurable and such tha
\[
P(F)\geq\frac{1}{n^{2}}(1-\varepsilon).
\]
Furthermore, for any $0\leq i,j\leq n-1,$ $0\leq k,\ell\leq n-1$ with
$(i,j)\neq(k,\ell)$ we have
\begin{equation}
P(T_{i,j}^{-1}F\cap T_{k,l}^{-1}F)=0. \label{disjoint
\end{equation}

\end{lemma}

\textbf{Proof of Lemma \ref{Rohling2}}.

\bigskip

Let $n$ be an integer and let $\varepsilon>0.$ By using Rokhlin lemma (see
Theorem \ref{Rohling} in Section \ref{AUX}), construct $B\in\mathcal{K}$ with
\begin{equation}
P(B)\geq(1-\frac{\varepsilon}{2})\frac{1}{n^{2}} \label{estB
\end{equation}
and for $0\leq i,j\leq n-1$, $T_{i,j}^{-1}B$ are disjoint for distinct pair of
indexes. Since $\mathcal{K}$ is generated by the field $\cup_{\mathbf{n
}\mathcal{F}_{\mathbf{n}},$ we can find a set $E$ in $\cup_{\mathbf{n
}\mathcal{F}_{\mathbf{n}}$ such that
\begin{equation}
P(B\Delta E)<\frac{\varepsilon}{8n^{4}}. \label{sym
\end{equation}
Since $E$ belongs to $\cup_{\mathbf{n}}\mathcal{F}_{\mathbf{n}},$ there is a
$\mathbf{m}$ such that $E\in\mathcal{F}_{\mathbf{m}}.$ So $T_{\mathbf{m
}(E)\in\mathcal{F}_{\mathbf{0}}.$ Denote $G=T_{\mathbf{m}}(E)$ and se
\[
F=G\setminus\cup_{(i,j)\in D}T_{i,j}^{-1}G,
\]
where $D=\{0\leq i,j\leq n-1,(i,j)\neq(0,0)\}$. Note now that for all
$(i,j)\in D,
\[
P(F\cap T_{i,j}^{-1}F)=0,
\]
which implies (\ref{disjoint}). Also, by stationarity,
\[
P(F)=P(E)-P(E\cap(\cup_{(i,j)\in D}T_{i,j}^{-1}E))\geq P(E)-\sum
\nolimits_{(i,j)\in D}P(E\cap T_{i,j}^{-1}E).
\]
But for $(i,j)\in D,$
\[
P(E\cap T_{i,j}^{-1}E)\leq2P(E\setminus B)\leq\frac{\varepsilon}{4n^{4}}.
\]
Therefore, by the above considerations, (\ref{sym}) and (\ref{estB}) we obtai
\[
P(F)\geq P(E)-\frac{\varepsilon}{4n^{2}}\geq P(B)-\frac{\varepsilon}{8n^{4
}-\frac{\varepsilon}{4n^{2}}\geq\frac{1-\varepsilon}{n^{2}}.
\]
$\square$

\bigskip

Next, we obtain a lemma which is the main step in the construction of the
example. In the sequel, we use the notation $a_{n}\sim b_{n}$ for
$\lim_{n\rightarrow\infty}a_{n}/b_{n}=1.$

\begin{lemma}
\label{NTight}There is a strictly stationary random field of integrable
positive random variables $(U_{i,j})_{i,j\in Z},$ coordinatewise ergodic, such
that for any $0<\varepsilon<1,$ $E|U_{0,0}|\ln^{1-\varepsilon}(1+|U_{0,0
|)<\infty$ and such that for almost all $\omega,$ $(U_{n,v}/nv)_{n,v\in Z}$ is
not tight under $P^{\omega}.$
\end{lemma}

\textbf{Proof of Lemma \ref{NTight}}.

\bigskip

By Lemma \ref{Rohling2}, for $n\geq2$ and $\varepsilon=1/2,$ we can find sets
$F_{n}\in\mathcal{F}_{-n,-n}$ such that $P(F_{n})=1/2n^{2}$ and such that for
any $0\leq i,j\leq n-1,$ $0\leq k,\ell\leq n-1$ with $(i,j)\neq(k,\ell)$ we
have $P(T_{i,j}^{-1}F_{n}\cap T_{k,l}^{-1}F_{n})=0.$

Now, we consider independent copies of the probability space $(\Omega
,\mathcal{K},P),$ denoted by $(\Omega^{(m)},\mathcal{K}^{(m)},P^{(m)
)_{m\geq1},$ and introduce the product space~$\mathbf{\Omega}=\prod
_{m=1}^{\infty}\Omega^{(m)}$ endowed with the sigma algebra generated by
cylinders, $\mathbf{K}=\prod_{m=1}^{\infty}\mathcal{K}^{(m)}$. We also
introduce on $\mathbf{K}$ the product probability $\mathbf{P}=\prod
_{m=1}^{\infty}P^{(m)},$ $P^{(m)}=P$. In this space consider sets $F_{n
^{(n)}$ which are products of $\Omega$ with the exception of the $n$-th
coordinate which is $F_{n}.$

On $\mathbf{\Omega,}$ define a random variable $f_{n}$ by the following
formula:
\begin{equation}
f_{n}=\frac{n}{\ln^{2}n}1_{F_{n}^{(n)}.} \label{def fn
\end{equation}
Let $A_{n}$ be the following event:
\[
A_{n}=\{\text{there are }i,j\text{, }\ln n\leq i,j\leq n-1,\text{ such that
}f_{n}\circ\mathbf{T}_{i,j}/ij\geq1\}.
\]
where $\mathbf{T}_{i,j}=(T_{i,j},T_{i,j},...).$ Since $f_{n}\circ
\mathbf{T}_{i,j}$ is $\prod\nolimits_{m=1}^{\infty}\mathcal{F}_{0,0}^{(m)}$
measurable, for $\omega\in A_{n},$ there are $i,j$, $\ln n\leq i,j\leq n-1,$
such that
\begin{equation}
\mathbf{P}^{\omega}(f_{n}\circ\mathbf{T}_{i,j}/ij\geq1)=1. \label{basic
\end{equation}
Note now that $f_{n}\circ\mathbf{T}_{i,j}/ij\geq1$ if and only if
$1_{F_{n}^{(n)}}\circ\mathbf{T}_{i,j}\geq ij(\ln n)^{2}/n,$ if and only if
$\omega\in(\mathbf{T}_{i,j})^{-1}(F_{n}^{(n)})$ and $ij\leq n/(\ln n)^{2}$.

Then, the probability of $A_{n}$ can be computed as
\[
\mathbf{P}(A_{n})=\mathbf{P}(\bigcup\nolimits_{D}\mathbf{T}_{i,j}^{-1
(F_{n}^{(n)})=P(\bigcup\nolimits_{D}T_{i,j}^{-1}(F_{n})),
\]
where the union and have indexes in the set $D=\{ij\leq(n-1)/(\ln n)^{2};\ln
n\leq i,j\leq n-1\}.$ By Lemma \ref{Rohling2}, it follows tha
\[
\mathbf{P}(A_{n})=P(F_{n})\sum_{\ln n\leq j\leq n-1}\text{ \ \ }\sum_{\ln
n\leq i\leq(n-1)/j(\ln n)^{2}}1\sim\frac{n\ln n}{2n^{2}(\ln n)^{2}}=\frac
{1}{2n\ln n}.
\]
Therefor
\[
\sum_{n\geq2}\mathbf{P}(A_{n})=\sum_{n\geq2}\frac{1}{2n\ln n}=\infty.
\]
By the second Borel-Cantelli lemma, $\mathbf{P}(A_{n}$ i.o.$)=1$. This means
that almost all $\omega\in\mathbf{\Omega}$ belong to an infinite number of
$A_{n}$. Whence, taking into account (\ref{basic}), for almost all $\omega
\in\mathbf{\Omega}$ and every positive $B,
\begin{equation}
\lim\sup_{i\wedge j\rightarrow\infty}\mathbf{P}^{\omega}(f_{m}\circ
\mathbf{T}_{i,j}/ij\geq B)=1. \label{f untight
\end{equation}
Define no
\begin{equation}
U_{0,0}=\sum_{n\geq2}f_{n}\text{ \ and \ }U_{i,j}=\sum_{n\geq2}f_{n
\circ\mathbf{T}_{i,j}. \label{def U
\end{equation}
Let us estimate the Luxembourg norm of $U_{0,0}$ in the Orlicz space generated
by the convex function $g(x)=x\ln^{1-\varepsilon}(1+x)$ for $x>0$,
$0<\varepsilon<1$. For each $n\in N$
\[
||f_{n}||_{g}=\inf_{\lambda}\{\lambda:E(\frac{f_{n}}{\lambda}\ln
^{1-\varepsilon}(1+\frac{f_{n}}{\lambda}))\leq1\}\text{ }.
\]
By the definition of $f_{n}$, we have
\begin{align*}
E(\frac{f_{n}}{\lambda}\ln^{1-\varepsilon}(1+\frac{f_{n}}{\lambda}))  &
=P(F_{n})\frac{n}{\lambda\ln^{2}n}\ln^{1-\varepsilon}(1+\frac{n}{\lambda
\ln^{2}n})\\
&  =\frac{1}{2\lambda n\ln^{2}n}\ln^{1-\varepsilon}(1+\frac{n}{\lambda\ln
^{2}n}).
\end{align*}
From this identity we see that, after some computations, that for $n$
sufficiently larg
\[
||f_{n}||_{g}\leq\frac{1}{n\ln^{1+\varepsilon/2}n}.
\]
Clearly, we hav
\begin{equation}
||U_{0,0}||_{g}\leq\sum_{n\geq2}||f_{n}||_{g}<\infty. \label{orlicz norm
\end{equation}
It remains to note that, by definition (\ref{def U}), $U_{i,j}\geq f_{n
\circ\mathbf{T}_{i,j}$. Therefore, by (\ref{f untight}) we also have for
almost all $\omega\in\mathbf{\Omega}$ and every positive $B,
\[
\lim\sup_{i\wedge j\rightarrow\infty}\mathbf{P}^{\omega}(U_{i,j}/ij\geq B)=1
\]
and the conclusion of this lemma follows by letting $B\rightarrow\infty$.
$\square$

\bigskip

\textbf{End of proof of Theorem \ref{Counterex}}

\bigskip

On the space constructed in Lemma \ref{NTight} define the independent random
variables $\xi_{i,j}^{^{\prime}}(\omega_{1},\omega_{2},...)=\xi_{i,j
(\omega_{1})~$and the random variables $X_{i,j}=\xi_{i,j}^{\prime
U_{i-1,j-1}^{1/2},$ where $(U_{i,j})_{i,j\in Z}$ and $(\xi_{i,j})_{i,j\in Z}$
are as in Lemma \ref{NTight}. Note that $(X_{i,j})_{i,j\in Z}$ is a sequence
of orthomartingale differences with respect to $\mathbf{\ }\prod_{m=1
^{\infty}\mathcal{F}_{i,j}^{(m)}$, where $\mathcal{F}_{i,j}^{(m)}$ are
independent copies of $\mathcal{F}_{i,j}$. According to Lemma \ref{NTight} for
$P-$almost all $\omega\in\mathbf{\Omega}$ we have
\[
\lim_{B\rightarrow\infty}\lim\sup_{i\wedge j\rightarrow\infty}P^{\omega
}(|X_{i,j}|/\sqrt{ij}\geq B)=1.
\]
If we assume now that $(S_{n,m}/\sqrt{nm})_{n,m\geq1}$ satisfies the quenched
limit theorem (or it is "quenched" tight), becaus
\[
U_{i-1,j-1}^{1/2}=|X_{i,j}|\leq|S_{i,j}|+|S_{i-1,j}|+|S_{i,j-1}|+|S_{i-1,j-1
|,
\]
then necessarily the field $(|X_{m,m}|/\sqrt{nm})_{n,m\geq1}$ should be tight
under $P^{\omega},$ for almost all $\omega$, which leads to a contradiction.
Note that, by (\ref{orlicz norm}), for any $0<\varepsilon<1$ we have
$EX_{0,0}^{2}\ln^{1-\varepsilon}(1+|X_{0,0}|)<\infty.$ For this example
$EX_{0,0}^{2}\ln(1+|X_{0,0}|)=\infty,$ since otherwise the quenched result
follows by Theorem \ref{QCLTmart}. $\square$

\section{Quenched functional CLT \label{functional CLT}}

In this section we formulate the functional CLT, which holds under the same
conditions as in Theorem \ref{QCLTmart}. For $(s,t)\in\lbrack0,1]^{2},$ we
introduce the stochastic process
\[
W_{n,v}(t,s)=\frac{1}{\sqrt{nv}}S_{[nt],[vs]}.
\]
We shall establish the following result. Denote by $(W(t,s))_{(t,s)\in
\lbrack0,1]^{2}}$ the standard $2$-dimensional Brownian sheet.

\begin{theorem}
\label{Th funct CLT}Under the setting of Theorem \ref{QCLTmart}, if we assume
that $E(X_{0,0}^{2})<\infty$ then, for $P$-almost all $\omega,$ the sequence
of processes $(W_{n,n}(t,s))_{n\geq1}$ converges in distribution in
$D([0,1]^{2})$ endowed with the uniform topology to $\sigma W(t,s),$ under
$P^{\omega}$. If we assume now that (\ref{moment condition}) holds, then for
$P$-almost all $\omega,$ the sequence $(W_{n,v}(t,s))_{n,v\geq1}$ converges in
distribution to $\sigma W(t,s),$ as $n\wedge v\rightarrow\infty$ under
$P^{\omega}$.
\end{theorem}

\textbf{Proof of Theorem \ref{Th funct CLT}}

\bigskip

Let us first prove the second case, when $n\wedge v\rightarrow\infty.$ As
usual, the proof of this theorem involves two steps, namely the proof of the
convergence of the finite dimensional distributions to the corresponding ones
of the standard $2$-dimensional Brownian sheet and tightness.

For proving tightness we shall verify the moment condition given in relation
(3) in Bickel and Wichura (1971) and then the tightness follows from Theorem 3
in the same paper. To verify it is enough to compute the $4-$th moment of an
increment of the process $W_{n,v}(t,s)$ on the rectangle $A=[t_{1
,t_{2})\times\lbrack s_{1},s_{2}).$ That is $E(\Delta^{4}(A))$ where
\[
\Delta(A)=\frac{1}{\sqrt{nv}}\sum\nolimits_{i=[nt_{1}]}^{[nt_{2}]-1
\sum\nolimits_{j=[vs_{1}]}^{[vs_{2}]-1}X_{i,j}.
\]
By applying Burkholder's inequality twice consecutively, and taking into
account that the variables are bounded by $C,$ for a positive constant $K$ we
obtain
\[
E^{\omega}(\Delta^{4}(A))\leq KC^{4}(t_{2}-t_{1})^{2}(s_{2}-s_{1})^{2
=KC^{4}\mu^{2}(A),
\]
where $\mu$ is the Lebesgue measure on $[0,1]^{2}.$ If $B$ is a neighboring
rectangle of $A$, by the Cauchy-Schwatz inequality we have
\[
E^{\omega}(\Delta^{2}(A)\Delta^{2}(B))\leq KC^{4}\mu(A)\mu(B).
\]
Therefore the moment condition in relation (3) in Bickel and Wichura (1971) is
verified with $\gamma=4$ and $\beta=2.$

The proof of the convergence of finite dimensional distribution follows, up to
a point, the proof of the corresponding result in Cuny et al. (2015), which
will be combined with the method of proof in Theorem \ref{QCLTmart}. As
explained in Subsection 3.2 in Cuny et al. (2015), in order to establish the
convergence of the finite dimensional distributions, we have to show that for
$P-$almost all $\omega\in\Omega,$ and for any partitions $0\leq t_{1
\leq...\leq t_{K}\leq1$ and $0\leq s_{1}\leq...\leq s_{K}\leq1,$ we hav

\begin{equation}
\frac{1}{\sqrt{nv}}\sum\nolimits_{k=1}^{K}\sum\nolimits_{\ell=1}^{K}a_{k,\ell
}\sum\nolimits_{i=[nt_{k-1}]}^{[nt_{k}]-1}\sum\nolimits_{j=[vs_{\ell-1
]}^{[vs_{\ell}]-1}X_{i,j}\Rightarrow N(0,\Gamma)\text{ under }P^{\omega
},\text{ } \label{fdd
\end{equation}
where $\Gamma=\sigma^{2}\sum\nolimits_{k=1}^{K}\sum\nolimits_{\ell=1
^{K}a_{k,\ell}^{2}(t_{k}-t_{k-1})(s_{\ell}-s_{\ell-1}).$ Since we have proved
tightness in $C([0,1]^{2}),$ we know that any subsequence contains one which
is converges in distribution to a continuous process. Therefore, without
restricting the generality we can restrict ourselves to partitions with
rational ends which form a countable set.

In order to establish this weak convergence we follow step by step the proof
of Theorem \ref{QCLTmart}. We shall just mention the differences. The first
step is to decompose $X_{i,j}$ as in formula (\ref{truncation}) and to show
the negligibility of the term containing $X_{i,j}^{"}.$ This is the only step
where we need different moment conditions according to whether indexes in the
sum are restricted or not. By using simple algebraic manipulations, the
triangle inequality along with Theorem 3.2\ in Billingsley (1999), we can
easily see that this term is negligible $P$-a.s. for the convergence in
$D([0,1]^{2})$ endowed with the uniform topology, if, for every $\varepsilon
>0$
\[
\lim_{A\rightarrow\infty}\text{ }\lim\sup_{n\wedge v\rightarrow\infty
P_{0,0}(\max_{1\leq i\leq n}\max_{1\leq j\leq v}|\sum\nolimits_{k=1}^{i
\sum\nolimits_{\ell=1}^{j}\mathcal{P}_{k,\ell}(X_{k,\ell}^{"})|>\varepsilon
\sqrt{nv})=0\text{ a.s.
\]
But by using Cairoli's maximal inequality for orthomartinagles (see Theorem
2.3.1 in Khoshnevisan, 2002, p. 19) the proof is reduced to showing
(\ref{negli1}), which was already established in proof of Theorem
\ref{QCLTmart}. Without loss of generality we redenote $\mathcal{P
_{i,j}(X_{i,j}^{\prime})$ by $X_{i,j}$ and assume that it is bounded by a
positive constant $C$. We continue the steps of the proof in Theorem
\ref{QCLTmart} and we shall verify the conditions of Theorem \ref{cltGH} with
the exception that we replace $F_{i,v}$ in definition (\ref{defFiv}) b
\[
F_{k,i,v}=\frac{1}{\sqrt{v}}\sum\nolimits_{\ell=1}^{K}a_{k,\ell
\sum\nolimits_{j=[vs_{\ell-1}]}^{[vs_{\ell}]-1}X_{i,j},
\]
where $[nt_{k-1}]\leq i\leq\lbrack nt_{k}]-1;$ $1\leq k\leq K.$ We also
replace $\sigma^{2}$ by $\eta_{k}^{2}=\sigma^{2}\sum\nolimits_{\ell=1
^{K}a_{k,\ell}^{2}(s_{\ell}-s_{\ell-1})$ and $h_{m,N_{0}}$ by
\[
h_{k,m,N_{0}}=\sup_{v>N_{0}}E_{0,0}|\frac{1}{m}\sum\nolimits_{i=0
^{m-1}F_{k,i,v}^{2}-\eta_{k}^{2}|.
\]
For instance, let us convince ourselves that (\ref{erg2}) holds. Indeed by the
ergodicity of $S$ and the fact that the variables are bounde
\[
\lim_{v\rightarrow\infty}E_{\infty,0}F_{k,i,v}^{2}=\lim_{v\rightarrow\infty
}\frac{1}{v}E_{\infty,0}(\sum\nolimits_{\ell=1}^{K}a_{k,\ell}\sum
\nolimits_{j=[vs_{\ell-1}]}^{[vs_{\ell}]-1}X_{i,j}^{2})=\eta_{k}^{2}.
\]

After we verify the conditions of Theorem \ref{cltGH} for the triangular array
of martingale differences $(F_{k,i,v})_{[nt_{k-1}]\leq i\leq\lbrack
nt_{k}]-1;\text{ }1\leq k\leq K}$ , we obtain the result in (\ref{fdd}) by
applying the CLT in Theorem \ref{cltGH}. \ $\square$

\section{Quenched functional CLT via coboundary decomposition \label{cob}}

Now we indicate a larger class than the orthomartingale, which satisfies a
quenched functional CLT. A fruitful approach is to approximate $S_{m,n}$ by an
orthomartingale $M_{n,m}$ in a norm that makes possible to transport the
quenched functional CLT given in Theorem \ref{Th funct CLT}. Such an
approximation is of the form: for every $\varepsilon>0,
\begin{equation}
\lim\sup_{n\wedge v\rightarrow\infty}P^{\omega}(\max_{1\leq k\leq n,1\leq
\ell\leq v}|S_{k,\ell}-M_{k,\ell}|>\varepsilon\sqrt{nv})=0\text{
}\ \text{a.s.} \label{ortho approx
\end{equation}

The random fields we consider can be decomposed into an orthomartingale and a
generalized coboundary and therefore satisfy (\ref{ortho approx}). This type
of orthomartingale approximation, so called martingale-coboundary
decomposition, was introduced for random fields by Gordin (2009) and studied
by El Machkouri and Giraudo (2016), Giraudo (2018) and Voln\'{y} (2018).

\begin{definition}
\label{defcobdec}We say that a random field $(X_{i,j})_{i,j\in Z}$, defined by
(\ref{def X}),\ adapted to the commuting filtration $(\mathcal{F
_{i,j})_{i,j\in Z},$ defined by (\ref{def F}),\ admits a martingale-coboundary
decomposition i
\begin{equation}
X_{0,0}=m_{0,0}+(1-\hat{T})m_{0,0}^{\prime}+(1-\hat{S})m_{0,0}^{"}+(1-\hat
{T})(1-\hat{S})Y_{0,0}, \label{decomp
\end{equation}
with $m_{0,0}$ an orthomartingale difference (satisfying (\ref{ortho})),
$m_{0,0}^{\prime}$ a martingale difference in the second coordinate and
$m_{0,0}^{"}$ a martingale difference in the first coordinate. All these
functions are $\mathcal{F}_{0,0}-$measurable.
\end{definition}

We shall obtain the following generalization of Theorem \ref{Th funct CLT}:

\begin{theorem}
\label{CoQCLT}Let us assume that the decomposition (\ref{decomp}) holds with
all the variables square integrable and $S$ $($or $T)\ $is ergodic. Then for
almost all $\omega\in\Omega,
\begin{equation}
\frac{1}{n}S_{[nt],[ns]}\Rightarrow|c|W(t,s)\text{ under }P^{\omega}\text{
when }n\rightarrow\infty, \label{clt1
\end{equation}
where $(W(t,s))_{(t,s)\in\lbrack0,1]^{2}}$ is the standard $2$-dimensional
Brownian sheet and $c^{2}=E(m_{0,0}^{2})$. If we assume that all the variables
involved in the decomposition (\ref{decomp}) satisfy (\ref{moment condition})
then, for almost all $\omega\in\Omega,$
\begin{equation}
\frac{1}{(nv)^{1/2}}S_{[nt],[vs]}\Rightarrow|c|W(t,s)\text{ under }P^{\omega
}\text{ when }n\wedge v\rightarrow\infty. \label{CLT 2
\end{equation}

\end{theorem}

It should be noted that Giraudo (2018) have shown that if
\begin{equation}
\sup_{n,v\geq0}E((E_{0,0}(S_{n,v}))^{2})<\infty, \label{con1
\end{equation}
then the decomposition (\ref{decomp}) holds and all the variables are in
$L_{2}.$ As a matter of fact this is also a necessary condition for
(\ref{decomp}). The only condition specific to $L_{2}$ needed for his proof is
the reflexivity of $L_{2}.$ Since the Orlicz space $L_{\varphi}$ generated by
the function
\[
\varphi(x)=x^{2}\log(1+x):[0,\infty)\rightarrow\lbrack0,\infty)
\]
is reflexive (see Theorem 8 in Milnes (1957)), the proof of Theorem 2.1 in
Giraudo is also valid in this context. It follows that if
\begin{equation}
\sup_{n,v\geq0}E(\varphi(|E_{00}(S_{n,v})|))<\infty, \label{con2
\end{equation}
then the decomposition in (\ref{decomp}) holds all the functions are in
$L_{\varphi}.$ The reciprocal is also true.

As a matter of fact, by combining Theorem \ref{CoQCLT} with this result we
deduce the following corollary:

\begin{corollary}
\label{corco}Let us assume that the random field $(X_{i,j})_{i,j\in Z}$,
defined by (\ref{def X}),\ adapted to the commuting filtration $(\mathcal{F
_{i,j})_{i,j\in Z},$ defined by (\ref{def F}),~satisfies (\ref{con1}). Then
$\lim_{n\wedge v\rightarrow\infty}(nv)^{-1}E(S_{n,v}^{2})=c^{2}$. If in
addition we assume that $S$ $($or $T)\ $is ergodic, then for $P-$almost all
$\omega\in\Omega,$ (\ref{clt1}) holds. Also, if condition (\ref{con2}) is
satisfied, then for $P-$almost all $\omega\in\Omega,$ (\ref{CLT 2}) holds.
\end{corollary}

\bigskip

\textbf{Proof of Theorem \ref{CoQCLT}}

\bigskip

Consider first that the indexes $n$ and $m$ are varying independently. Denote
by $m_{i,j}=m_{0,0}\circ T_{i,j}$ and $M_{k,\ell}=\sum_{i=0}^{k-1}\sum
_{j=0}^{\ell-1}m_{i,j}.$

We shall establish (\ref{ortho approx}). A simple computation shows that
$(S_{k,\ell}-M_{k,\ell})/\sqrt{nv}$ is the sum of the following three terms
\[
\frac{1}{\sqrt{nv}}\sum_{i=0}^{k-1}\sum_{j=0}^{\ell-1}\hat{T}^{i}\hat{S
^{j}(I-\hat{T})m_{0,0}^{\prime}=\frac{1}{\sqrt{nv}}\sum_{j=0}^{\ell-1}\hat
{S}^{j}(m_{0,0}^{\prime}-\hat{T}^{k}m_{0,0}^{\prime})=R_{1}(k,\ell),
\
\[
\frac{1}{\sqrt{nv}}\sum_{i=0}^{k-1}\sum_{j=0}^{\ell-1}\hat{T}^{i}\hat{S
^{j}(I-\hat{S})m_{0,0}^{"}=\frac{1}{\sqrt{nv}}\sum_{i=0}^{k-1}\hat{T
^{i}(m_{0,0}^{"}-\hat{S}^{\ell}m_{0,0}^{"})=R_{2}(k,\ell),
\
\[
\frac{1}{\sqrt{nv}}\sum_{i=0}^{k-1}\sum_{j=0}^{\ell-1}\hat{T}^{i}\hat{S
^{j}(I-\hat{T})(I-\hat{S})Y_{0,0}=\frac{1}{\sqrt{nv}}(I-\hat{S}^{\ell
)(I-\hat{T}^{k})Y_{0,0}=R_{3}(k,\ell).
\]
In order to treat the last term, note tha
\[
\max_{1\leq k\leq n,1\leq\ell\leq v}|R_{3}(k,\ell)|\leq\frac{4}{\sqrt{nv
}\text{ }\max_{0\leq i\leq n}\text{ }\max_{0\leq j\leq v}|Y_{i,j}|.
\]
Let $A$ be a positive integer. By truncation at the level $A$ we obtain the
following boun
\[
\frac{1}{nv}\max_{0\leq i\leq n}\max_{0\leq j\leq v}|Y_{i,j}|^{2}\leq
\frac{A^{2}}{nv}+\frac{1}{nv}\sum_{i=0}^{n}\sum_{j=0}^{v}Y_{i,j}^{2
I(|Y_{i,j}|>A).
\]
Because of the stationarity and the fact that in the second part of Theorem
\ref{CoQCLT}\textbf{ }we imposed condition\textbf{ (}\ref{moment condition
\textbf{)}, by the ergodic theorem for stationary random fields (see Theorem
1.1 in Ch.6, Krengel (1985)) it follows that for every $A,$
\[
\lim_{n\wedge v\rightarrow\infty}\frac{1}{nv}\sum_{i=0}^{n}\sum_{j=0
^{v}Y_{i,j}^{2}I(|Y_{i,j}|>A)=E(Y_{0,0}^{2}I(|Y_{0,0}|>A)).
\]
Therefore $\lim_{A\rightarrow\infty}\lim_{n\wedge v\rightarrow\infty
|R_{3}(n,v)|=0$ $P-$a.s. By Fubini's theorem, it follows that the limit is $0$
also under $P^{\omega}$, for almost all $\omega$.

The terms $R_{1}(k,\ell)$ and $R_{2}(k,\ell)$ are treated similarly, with
small differences. Let us treat the first one only. It is convenient to
truncate at a positive number $A$.$\ $Le
\begin{align*}
m_{j,k}^{\prime}  &  =m_{j,k}^{\prime}I(|m_{j,k}^{\prime}|\leq A)-E_{j,k-1
m_{j,k}^{\prime}I(|m_{j,k}^{\prime}|\leq A)+\\
m_{j,k}^{\prime}I(|m_{j,k}^{\prime}|  &  >A)-E_{j,k-1}m_{j,k}^{\prime
}I(|m_{j,k}^{\prime}|>A).
\end{align*}
We shall use the following bound
\begin{gather*}
E_{0,0}\max_{1\leq k\leq n,1\leq\ell\leq v}R_{1}^{2}(k,\ell)\leq2E_{0,0
\max_{1\leq k\leq n,1\leq\ell\leq v}(\sum_{j=0}^{\ell-1}m_{j,k}^{\prime
)^{2}\leq\\
8A^{2}v+2E_{0,0}\max_{1\leq k\leq n,1\leq\ell\leq v}(\sum_{j=0}^{\ell
-1}m_{j,k}^{\prime}I(|m_{j,k}^{\prime}|>A)-E_{j,k-1}m_{j,k}^{\prime
I(|m_{j,k}^{\prime}|>A))^{2}\\
\leq8A^{2}v+2\sum_{k=1}^{n}E_{0,0}\max_{1\leq\ell\leq v}(\sum_{j=0}^{\ell
-1}m_{j,k}^{\prime}I(|m_{j,k}^{\prime}|>A)-E_{j,k-1}m_{j,k}^{\prime
I(|m_{j,k}^{\prime}|>A))^{2}.
\end{gather*}
Now, by the Doob's maximal inequality
\begin{gather*}
\frac{1}{nv}E_{0,0}\max_{1\leq k\leq n,1\leq\ell\leq v}R_{1}^{2}(k,\ell)\\
\leq\frac{8A^{2}}{n}+\frac{2}{nv}\sum_{k=1}^{n}\sum_{j=0}^{v-1}E_{0,0
(m_{j,k}^{\prime}I(|m_{j,k}^{\prime}|>A)-E_{j,k-1}m_{j,k}^{\prime
I(|m_{j,k}^{\prime}|>A))^{2}\\
\leq\frac{8A^{2}}{n}+\frac{4}{nv}\sum_{k=1}^{n}\sum_{j=0}^{v-1}E_{0,0
(m_{j,k}^{\prime}I(|m_{j,k}^{\prime}|>A))^{2}\\
=\frac{8A^{2}}{n}+\frac{4}{nv}\sum_{k=1}^{n}\sum_{j=0}^{v-1}Q_{1}^{j}Q_{2
^{k}[(m_{0,0}^{\prime})^{2}I(|m_{0,0}^{\prime}|>A)].
\end{gather*}
We let $n\wedge v\rightarrow\infty$\ and we use Theorem 1.1 in Ch. 6 of
Krengel (1985). It follows that, for every $A$
\[
\lim_{n\wedge v\rightarrow\infty}\frac{1}{nv}E_{0,0}\max_{1\leq k\leq
n,1\leq\ell\leq v}R_{1}^{2}(k,\ell)=E(m_{0,0}^{\prime})^{2}I(|m_{0,0}^{\prime
}|>A).
\]
Then, we let $A\rightarrow\infty.$ This completes the proof of
(\ref{ortho approx}). The result follows by using the second part of Theorem
\ref{Th funct CLT} along with Theorem 3.2 in Billingsley (1999). Now for the
situation $n=m\rightarrow\infty,$ the proof is similar with the difference
that we use Theorem 3.5 in Ch. 6 in Krengel (1985)\ instead of Theorem 1.1 in
the same chapter together with the first part of Theorem \ref{Th funct CLT}.
$\square$

\begin{remark}
If we take $Y_{0,0}$, in the martingale-coboundary decomposition
(\ref{decomp}),\ to be the function $U_{0,0}^{1/2}$ found in the proof of
Lemma \ref{NTight}, then for almost all $\omega$
\[
R_{3}(n,v)=\frac{1}{\sqrt{nv}}\sum_{i=0}^{n-1}\sum_{j=0}^{v-1}\hat{T}^{i
\hat{S}^{j}(I-\hat{T})(I-\hat{S})Y_{0,0
\]
does not converge to $0$ in probability $P^{\omega}$ when $n\wedge
v\rightarrow\infty.$ Therefore if only the existence of the second moment is
assumed or even if $EY_{0,0}^{2}\ln^{1-\varepsilon}(1+|Y_{0,0}|)<\infty$ for
some $0<\varepsilon<1,$ this coboundary could spoil the quenched weak
convergence. This is in sharp contrast with the dimension $1$. Recall that in
dimension $1$, when we have a martingale-coboundary decomposition $X_{0
=D_{0}+G_{0}-\hat{T}G_{0}$ with $D_{0}$ a martingale difference and $G_{0}\in
L_{2},$ then the coboundary $G_{0}-\hat{T}G_{0}$ does not spoil the quenched
invariance principle (see Theorem 8.1 in Borodin and Ibragimov (1994), which
is due to Gordin and Lifshits). In higher dimension, in general, we need
stronger moment conditions not only for martingale differences but also for
the cobounding function $Y_{0,0}$.
\end{remark}

\bigskip

\section{The case of d-indexed random field \label{section gend}}

In this section we formulate our results and indicate their proofs for random
fields indexed by $Z^{d}$ with $d>2.$ The proofs are based on induction
arguments. When we add on unrestricted $d$-dimensional rectangles the moment
conditions will depend on $d$. By $\mathbf{u=}(u_{1},u_{2},...,u_{d})$ we
denote elements of $Z^{d}$. Let us suppose that $\mathbf{T}=(T_{i})_{1\leq
i\leq d}$ are $d$ commuting, invertible, measure preserving transformations
from $\Omega$ to $\Omega$ and let $\mathcal{F}_{\mathbf{0}}$ be a sub-sigma
field of $\mathcal{K}$. For all $\mathbf{u}\in Z^{d}$ define $\mathcal{F
_{\mathbf{u}}=\mathbf{T}^{-\mathbf{u}}(\mathcal{F}_{\mathbf{0}}),$ where
$\mathbf{T}^{-\mathbf{u}}$ is the following composition of operators:
$\mathbf{T}^{-\mathbf{u}}=\prod\nolimits_{i=1}^{n}T_{i}^{-u_{i}}.$ Assume the
filtration is coordinatewise increasing and commuting, in the sense that for
any integrable variable we have $E_{\mathbf{u}}E_{\mathbf{a}}X=E_{\mathbf{a
\wedge\mathbf{u}}X,$ where $\mathbf{a}\wedge\mathbf{u}$ means coordinatewice
minimum and we used the notation $E_{\mathbf{u}}X=E(X|\mathcal{F}_{\mathbf{u
})$. We introduce the stationary field by starting with a $\mathcal{F
_{\mathbf{0}}-$measurable function $X_{\mathbf{0}}:\Omega\rightarrow R$ and
then define the random field $X_{\mathbf{k}}(\omega)=X_{\mathbf{0}
(\mathbf{T}^{\mathbf{k}}(\omega))=X_{\mathbf{0}}(T_{1}^{k_{1}}\circ...\circ
T_{d}^{k_{d}}).$ The operator $\mathbf{\hat{T}}$ is defined on $L_{2}$ as
$\mathbf{\hat{T}(}f\mathbf{)=}f\circ\mathbf{T.}$ For the filtration
$(\mathcal{F}_{\mathbf{u}})_{\mathbf{u}\in Z^{d}},$ defined as above,\ we call
the random field $(X_{\mathbf{u}})_{\mathbf{u\in Z}^{d}}$ \ orthomartingale
difference if $E(X_{\mathbf{u}}|\mathcal{F}_{\mathbf{i}})=0$ when at least one
coordinate of $\mathbf{i}$ is strictly smaller that the corresponding
coordinate of $\mathbf{u}$. We also use the notation $\mathbf{i
\leq\mathbf{u,}$ where the inequality is coordinatewise and $|\mathbf{n
|=n_{1}\cdot...\cdot n_{d}.$ Finally denote $S_{\mathbf{n}}
{\displaystyle\sum\nolimits_{\mathbf{0}\leq\mathbf{i}\leq\mathbf{n-1}}}
X_{\mathbf{i}}.$ In this context we have:

\begin{theorem}
\label{trmd}Assume that there is an integer $i$, $1\leq i\leq d$ such that
$T_{i}$ is ergodic and $X_{\mathbf{0}}$ is square integrable, $E(X_{\mathbf{0
}^{2})=\sigma^{2}$. Then, for $P-$almost all $\omega\in\Omega,
\[
\frac{1}{n^{d/2}}S_{(n,n,...,n)}\Rightarrow\sigma W(t_{1},...,t_{d})\text{
under }P^{\omega}\text{ when }n\rightarrow\infty\text{ }.
\]
In addition, if $E[X_{\mathbf{0}}^{2}\log^{d-1}(1+|X_{\mathbf{0}}|)]<\infty$,
then for almost all $\omega\in\Omega,$
\[
\frac{1}{|\mathbf{n}|^{1/2}}S_{\mathbf{(}n_{1},n_{2},...,n_{d})
\Rightarrow\sigma W(t_{1},...,t_{d})\text{ under }P^{\omega}\text{ when
\min_{1\leq i\leq d}n_{i}\rightarrow\infty.
\]

\end{theorem}

\begin{remark}
\label{remark d}Both Theorems \ref{Th funct CLT} and \ref{CoQCLT} as well as
Corollary \ref{corco} also hold for the multi-indexed random field
$(X_{\mathbf{u}})_{\mathbf{u\in Z}^{d}}$ defined above.
\end{remark}

We shall indicate how to prove these results by induction. We shall follow
step by step the proof of Theorem \ref{QCLTmart} with the following
differences. Without restricting the generality, let us assume that the
operator $T_{i}$ is ergodic for an $i$, $2\leq i\leq d.$ We define now the
$d$-dimensional projection operators. By using the commutative property of the
filtrations it is convenient to define

\[
\mathcal{P}_{\mathbf{u}}(X)=\mathcal{P}_{\mathbf{u}_{1}}\circ\mathcal{P
_{\mathbf{u}_{2}}\circ...\circ\mathcal{P}_{\mathbf{u}_{d}}(X),
\]
wher
\[
\mathcal{P}_{\mathbf{u}_{j}}(Y)=E(Y|\mathcal{F}_{\mathbf{u}})-E(Y|\mathcal{F
_{\mathbf{u}_{j}}).
\]
Above we used the notation $\mathbf{u}_{j}$ for a vector which has the same
coordinates as $\mathbf{u}$ with the exception of the $j$-th coordinate, which
is $u_{j}-1$. For instance when $d=3,$ $\mathcal{P}_{\mathbf{u}_{2
}(Y)=E(Y|\mathcal{F}_{u_{1},u_{2},u_{3}})-E(Y|\mathcal{F}_{u_{1},u_{2
-1,u_{3}}).$~We can easily see that, by using the commutativity property of
the filtration, this definition is a generalization of the case $d=2$. We note
that, by using this definition of $\mathcal{P}_{\mathbf{u}}(X),$ the
truncation argument in Theorem \ref{QCLTmart} remains unchanged if we replace
the index set $Z^{2}$ with $Z^{d}.$ We point out the following two differences
in the proof of Theorem \ref{trmd}. One difference is that, for the validity
of the limit in (\ref{erg}) when $\min_{1\leq i\leq d}n_{i}\rightarrow\infty,$
in order to apply the ergodic theorem for Dunford-Schwartz operators, conform
to Ch. 6 Theorem 2.8 and Theorem 1.1 in Krengel (1985), we have to assume that
$E[X_{0,0}^{2}\log^{d-1}(1+|X_{0,0}|)]<\infty$. After we reduce the problem to
the case of bounded random variables, we proceed with the proof of the CLT by
induction. More precisely, we write the sum in the for
\[
\frac{1}{|\mathbf{n}|^{1/2}}S_{\mathbf{(}n_{1},n_{2},...,n_{d})}=\frac
{1}{n_{1}^{1/2}}\sum_{k_{1}=0}^{n_{1}-1}F_{k_{1},(n_{2},n_{3},...,n_{d
)}\text{ ,
\]
wit
\[
F_{k_{1},(n_{2},n_{3},...,n_{d})}=\frac{1}{(n_{2}\cdot...\cdot n_{d})^{1/2
}\sum_{\mathbf{k\in}B}X_{\mathbf{k}}\text{ ,
\]
where the sum is taken on the set $B=\{(0,...,0)\leq(k_{2}...k_{d})\leq
(n_{2}-1,...,n_{d}-1)\}.$ Because one operator is ergodic, according to the
induction hypothesis, $F_{k,(n_{2},n_{3},...,n_{d})}\Rightarrow N(0,\sigma
^{2})$ under $P^{\omega}$ for almost all $\omega,$ and we can replace
(\ref{erg2}) by
\[
\lim\frac{1}{n_{2}...n_{d}}E_{\infty,0,...0}\sum_{B}X_{\mathbf{k}}^{2
=\sigma^{2}\text{ a.s. when }\min(n_{2},...,n_{d})\rightarrow\infty.
\]

\section{Examples\label{Appl}}

We shall give examples providing new results for linear and Volterra random
fields with i.i.d. innovations. Let $d$ be an integer $d>1.$ Denote by
$\mathbf{t=(}t_{1},t_{2},...,t_{d})$ and let $W(\mathbf{t})$ be the standard
$d$-dimensional Brownian sheet.

\begin{example}
\label{exlinear} Let $(\xi_{\mathbf{n}})_{\mathbf{n}\in Z^{d}}$ be a random
field of independent, identically distributed random variables, which are
centered and have finite second moment. Let $(a_{\mathbf{n}})_{\mathbf{n}\in
Z^{d}}$ be a sequence of real numbers such that $\sum_{\mathbf{j
\geq\mathbf{0}}a_{\mathbf{j}}^{2}<\infty$. Defin
\[
X_{\mathbf{k}}=\sum_{\mathbf{j}\geq\mathbf{0}}a_{\mathbf{j}}\xi_{\mathbf{k
-\mathbf{j}}.
\]
Assume that
\begin{equation}
\sup_{\mathbf{n}\geq\mathbf{1}}\sum_{\mathbf{i}\geq\mathbf{0}}b_{\mathbf{n
,\mathbf{i}}^{2}<\infty,\text{ where }b_{\mathbf{n,i}}=\sum_{\mathbf{0
\leq\mathbf{k\leq n-1}}a_{\mathbf{k}+\mathbf{i}}. \label{lin
\end{equation}
Then, if $\mathbf{n}=(n,n,...,n),$ for $P-$almost all $\omega$
\begin{equation}
\frac{1}{n^{d/2}}S_{[(\mathbf{n-1)\cdot t]}}\Rightarrow|c|W(\mathbf{t})\text{
under }P^{\omega}\text{ when }n\rightarrow\infty. \label{ex1
\end{equation}
If we assume now that $E(\xi_{\mathbf{0}}^{2}\log^{d-1}(1+|\xi_{\mathbf{0
}|))<\infty,$ then for $P-$almost all $\omega$
\begin{equation}
\frac{1}{|\mathbf{n}|^{1/2}}S_{[(\mathbf{n-1)\cdot t]}}\Rightarrow
|c|W(\mathbf{t})\text{ under }P^{\omega}\text{ when }\min(n_{1},...,n_{d
)\rightarrow\infty, \label{ex2
\end{equation}
where $\mathbf{n}=(n_{1},...,n_{d})$.
\end{example}

\textbf{Proof of Example \ref{exlinear}}.

\bigskip

For this case we take $\mathcal{F}_{\mathbf{n}}=\sigma(\xi_{\mathbf{u
},\mathbf{u}\leq\mathbf{n}).$ Let us note first that the variables are square
integrable and well defined. We also have
\[
E(S_{\mathbf{n}}|\mathcal{F}_{\mathbf{0}})=\sum_{\mathbf{0}\leq\mathbf{k\leq
n-1}}\sum_{\mathbf{j}\leq\mathbf{0}}a_{\mathbf{k}-\mathbf{j}}\xi_{\mathbf{j}
\]
and therefor
\[
E(E^{2}(S_{\mathbf{n}}|\mathcal{F}_{\mathbf{0}}))=\sum_{\mathbf{i
\geq\mathbf{0}}(\sum_{\mathbf{0}\leq\mathbf{k\leq n-1}}a_{\mathbf{k
+\mathbf{i}})^{2}E(\xi_{\mathbf{1}}^{2}).
\]
The result follows for $S_{\mathbf{n}}$ by applying the first part of
Corollary \ref{corco}.

On the other hand, by the Rosenthal inequality for independent random
variables (see relation 21.5 in Burkholder (1973)), applied with the function
$\varphi(x)=x^{2}\log^{d-1}(1+|x|),$ there is a positive constant $C$ such
that
\[
E(\varphi(|E(S_{\mathbf{n}}|\mathcal{F}_{\mathbf{0}})|))\leq C\varphi\left(
\left(  \sum_{\mathbf{i}\geq\mathbf{0}}b_{\mathbf{n,i}}^{2}E(\xi_{\mathbf{1
}^{2})\right)  ^{1/2}\right)  +C\sum_{\mathbf{i}\geq\mathbf{0}}E(\varphi
(|b_{\mathbf{n,i}}\xi_{\mathbf{0}}|)),
\]
which is bounded under condition (\ref{lin}). Indeed, condition (\ref{lin})
implies that $\sup_{\mathbf{n}\geq\mathbf{1}}\sup_{\mathbf{i}\geq\mathbf{0
}|b_{\mathbf{n},\mathbf{i}}|<\infty,$ and then, after simple algebraic
manipulations we can find a positive constant $K$ such that
\[
E(\varphi(|b_{\mathbf{n,i}}\xi_{\mathbf{0}}|))\leq Kb_{\mathbf{n,i}
^{2}(E(\varphi(|\xi_{\mathbf{0}}|))+E(\xi_{\mathbf{0}}^{2})).
\]
It remains to apply the second part from Corollary \ref{corco} and Remark
\ref{remark d} in order to obtain the second part of the example. $\square$

\bigskip

Another class of nonlinear random fields are the Volterra processes, which
play an important role in the nonlinear system theory.

\begin{example}
\label{Volterra}Let $(\xi_{\mathbf{n}})_{\mathbf{n}\in Z^{d}}$ be a random
field of independent random variables, identically distributed, centered and
with finite second moment. Defin
\[
X_{\mathbf{k}}=\sum_{(\mathbf{u},\mathbf{v)}\geq(\mathbf{0},\mathbf{0
)}a_{\mathbf{u},\mathbf{v}}\xi_{\mathbf{k-u}}\xi_{\mathbf{k-v}},
\]
where $a_{\mathbf{u},\mathbf{v}}$ are real coefficients with $a_{\mathbf{u
,\mathbf{u}}=0$ and $\sum_{\mathbf{u,v}\geq\mathbf{0}}a_{\mathbf{u,v}
^{2}<\infty$. Denot
\[
c_{\mathbf{u},\mathbf{v}}(\mathbf{j})=\sum\limits_{\mathbf{0}\leq\mathbf{k\leq
j-1}}a_{\mathbf{k+u,k+v}}\text{.
\]
Assume tha
\begin{equation}
\sup_{\mathbf{j\geq1}}\sum\limits_{\mathbf{u}\geq\mathbf{0,v}\geq
\mathbf{0,u}\neq\mathbf{v}}c_{\mathbf{u},\mathbf{v}}^{2}(\mathbf{j})<\infty.
\label{volt
\end{equation}
Then the quenched functional CLT in (\ref{ex1}) holds. If in addition we
assume that $E(\xi_{\mathbf{0}}^{2}\log^{d-1}(1+|\xi_{\mathbf{0}}|))<\infty$,
then the quenched functional CLT in (\ref{ex2}) holds for sums of variables in
a general $d$-dimensional rectangle.
\end{example}

\textbf{Proof of Example \ref{Volterra}}.

\bigskip

For this case we consider the sigma algebras as in Example \ref{exlinear}. We
start from the following estimate
\[
E(S_{\mathbf{j}}|\mathcal{F}_{\mathbf{0}})=\sum_{(\mathbf{u},\mathbf{v)
\geq(\mathbf{0},\mathbf{0})}\sum\limits_{\mathbf{0}\leq\mathbf{k\leq j-1
}a_{\mathbf{k+u},\mathbf{k+v}}\xi_{-\mathbf{u}}\xi_{-\mathbf{v}
=\sum_{(\mathbf{u},\mathbf{v)}\geq(\mathbf{0},\mathbf{0})}c_{\mathbf{u
,\mathbf{v}}(\mathbf{j})\xi_{-\mathbf{u}}\xi_{-\mathbf{v}}.
\]
Since by our conditions $c_{\mathbf{u},\mathbf{u}}=0,$ by Tonelli theorem we
obtain
\begin{align*}
E(E^{2}(S_{\mathbf{j}}|\mathcal{F}_{\mathbf{0}}))  &  =\sum\limits_{\mathbf{u
\geq\mathbf{0,v}\geq\mathbf{0,u}\neq\mathbf{v}}(c_{\mathbf{u},\mathbf{v}
^{2}(\mathbf{j})+c_{\mathbf{u},\mathbf{v}}(\mathbf{j})c_{\mathbf{v
,\mathbf{u}}(\mathbf{j}))E(\xi_{\mathbf{u}}\xi_{\mathbf{v}})^{2}\\
&  \leq2\sum\limits_{\mathbf{u}\geq\mathbf{0,v}\geq\mathbf{0,u}\neq\mathbf{v
}(c_{\mathbf{u},\mathbf{v}}^{2}(\mathbf{j})+c_{\mathbf{v},\mathbf{u}
^{2}(\mathbf{j}))E(\xi_{\mathbf{u}}\xi_{\mathbf{v}})^{2}\\
&  \leq4\sum\limits_{\mathbf{u}\geq\mathbf{0,v}\geq\mathbf{0,u}\neq\mathbf{v
}c_{\mathbf{u},\mathbf{v}}^{2}(\mathbf{j})(E(\xi_{\mathbf{0}}^{2}))^{2}.
\end{align*}
The first result of this theorem follows by applying the first part of
Corollary \ref{corco} via Remark \ref{remark d}.

On the other hand, by a moment inequality for $U$-statistics based on the
decoupling procedures, (see Relation 3.1.3. in Gin\'{e} et al., 2000), we
obtain for $\varphi(x)=x^{2}\log^{d-1}(1+|x|),$
\[
E(\varphi(|E(S_{\mathbf{j}}|\mathcal{F}_{\mathbf{0}})|))\leq CE\varphi\left(
\sum_{(\mathbf{u},\mathbf{v)}\geq(\mathbf{0},\mathbf{0})}c_{\mathbf{u
,\mathbf{v}}(\mathbf{j})\xi_{-\mathbf{u}}\xi_{-\mathbf{v}}^{\prime}\right)  ,
\]
where $(\xi_{\mathbf{n}}^{\prime})_{\mathbf{n}\in Z^{d}}$ in an independent
copy of $(\xi_{\mathbf{n}})_{\mathbf{n}\in Z^{d}}$ and $C$ is a positive
constant. Now, we apply Rosenthal inequality, given in relation 21.5 in
Burkholder (1973), and we find a constant $C^{\prime}>0$ such that
\begin{align*}
E(\varphi(|E(S_{\mathbf{j}}|\mathcal{F}_{\mathbf{0}})|))  &  \leq C^{\prime
}\varphi\left(  \left(  \sum\limits_{\mathbf{u}\geq\mathbf{0,v}\geq
\mathbf{0,u}\neq\mathbf{v}}c_{\mathbf{u},\mathbf{v}}^{2}(\mathbf{j})E^{2
(\xi_{\mathbf{1}}^{2})\right)  ^{1/2}\right) \\
&  +C^{\prime}\sum_{(\mathbf{u},\mathbf{v)}\geq(\mathbf{0},\mathbf{0
)}E\varphi\left(  |c_{\mathbf{u},\mathbf{v}}(\mathbf{j})\xi_{-\mathbf{u}
\xi_{-\mathbf{v}}^{\prime}|\right)  .
\end{align*}
Note that, by (\ref{volt}), we have $\sup_{\mathbf{u},\mathbf{v\geq0,j\geq1
}|c_{\mathbf{u},\mathbf{v}}(\mathbf{j})|<\infty.$ Also, because $\xi
_{-\mathbf{u}}$ and $\xi_{-\mathbf{v}}^{\prime}$ are independent and
identically distributed, by the properties of $\varphi,$ we can find positive
constants such that
\begin{align*}
&  E\varphi\left(  |c_{\mathbf{u},\mathbf{v}}(\mathbf{j})\xi_{-\mathbf{u}
\xi_{-\mathbf{v}}^{\prime}|\right) \\
&  \leq Kc_{\mathbf{u},\mathbf{v}}^{2}(\mathbf{j})[E(\varphi(\xi_{\mathbf{0
}))E(\xi_{\mathbf{0}}^{2})+(E(\xi_{\mathbf{0}}^{2}))^{2}]\leq K^{\prime
}c_{\mathbf{u},\mathbf{v}}^{2}(\mathbf{j}).
\end{align*}
It remains to note that condition (\ref{volt}) implies condition (\ref{con2})
and then to apply the second part of Corollary \ref{corco} and Remark
\ref{remark d}.

\begin{remark}
In Examples \ref{exlinear} and \ref{Volterra} the innovations are i.i.d.
fields. However, the property (\ref{com}) for the filtration is not restricted
to filtrations generated by independent random variables. For example, we can
take as innovations the random field $(\xi_{n,m})_{n.m\in Z}$ having as
columns independent copies of a stationary and ergodic martingale differences
sequence. In this case the filtration generated $(\xi_{n,m})_{n,m\in Z}$ is
also commuting. As a matter of fact a commuting filtration could be generated
by a stationary random field $(\xi_{n,m})_{n,m\in Z}$ where the columns are
independent, i.e. $\bar{\eta}_{m}=(\xi_{n,m})_{n\in Z}$ are independent.
\end{remark}

\section{Auxiliary results \label{AUX}}

The following is a variant of Theorem 1 in G\"{a}nssler and H\"{a}usler (1979)
(see also G\"{a}nssler and H\"{a}usler, 1986, pages 315--317).

\begin{theorem}
\label{cltGH}Assume that $(D_{n,k})_{1\leq k\leq n}$ is a triangular array of
martingale differences adapted to an increasing filtration $(\mathcal{F
_{n,k})_{k}$. Assume that for all $q\ $rational numbers in $[0,1],
\begin{equation}
\sum_{k=1}^{[nq]}D_{n,k}^{2}\rightarrow^{P}\sigma^{2}q \label{GH
\end{equation}
and $\max_{1\leq k\leq n}|D_{n,k}|$ is uniformly integrable. Then
$S_{[nt]}\Rightarrow\sigma W(t),$ where $S_{[nt]}=\sum_{k=1}^{[nt]}D_{n,k}$
and $W(t)$ is a standard Brownian measure. In particular $S_{n}\Rightarrow
N(0,\sigma^{2})$.
\end{theorem}

As a matter of fact, condition (\ref{GH})\ in Theorem 1 in G\"{a}nssler and
H\"{a}usler (1979) is formulated for all reals $t\in\lbrack0,1].$ We notice
however that if (\ref{GH}) holds for any $q$ rational number in $[0,1]$ then
it also holds for any $t\in\lbrack0,1].$ To see it fix $t,$ $t\in\lbrack0,1]$
and let $q_{1}$ and $q_{2}$ be two rational numbers such that $q_{1}\leq t\leq
q_{2}.$ Then, by using monotonicity, note that
\[
\sum\nolimits_{k=1}^{[nq_{1}]}D_{n,k}^{2}-\sigma^{2}q_{2}\leq\sum
\nolimits_{k=1}^{[nt]}D_{n,k}^{2}-\sigma^{2}t\leq\sum\nolimits_{k=0
^{[nq_{2}]}D_{n,k}^{2}-\sigma^{2}q_{1
\]
and therefor
\[
|\sum\nolimits_{k=1}^{[nt]}D_{n,k}^{2}-\sigma^{2}t|\leq\max_{i=1,2
|\sum\nolimits_{k=0}^{[nq_{i}]}D_{n,k}^{2}-\sigma^{2}q_{i}|+(q_{2
-q_{1})\sigma^{2}.
\]
By using the hypothesis (\ref{GH}), and the fact that the rational numbers are
dense in $R$ it follows that (\ref{GH}) holds for any $t\in\lbrack0,1]$.

\bigskip

We mention now Theorem 34.2 (v) in Billingsley (1995). Further reaching
results including comments of the sharpness of the result below can be found
in Argiris and Rosenblatt (2006).

\begin{theorem}
\label{Bill}Assume that the sequence of random variables $(X_{n})_{n\geq0}$
converges a.s. to $X$ and there is an integrable and positive random variable
$Y$ such that $|X_{n}|\leq Y$ a.s. for all $n\geq0$. Let $\mathcal{F}$ be a
sigma algebra. Then the sequence\ $(E(X_{n}|\mathcal{F}))_{n\geq0}$ converges
a.s. to $E(X|\mathcal{F}).$
\end{theorem}

The following is a result in Katznelson and Weiss (1972) known under the name
of Rokhlin lemma for amenable actions\textbf{.}

\begin{theorem}
\textbf{\label{Rohling}} Let $(\Omega,\mathcal{K},P)$ be a nonatomic
probability space and let $T$ be a measure preserving action of $Z^{2}$ into
$\Omega$
\[
T:Z^{2}\times\Omega\rightarrow\Omega
\]
that is ergodic\textbf{.} Then, for all $\varepsilon>0$ and $n\in N$, there is
a set $B=B(n,\varepsilon)\in\mathcal{K}$ such that for $0\leq i,j\leq n-1$,
$T_{i,j}^{-1}B$ are disjoint for distinct indexes $(i,j)$ and
\[
P(B)\geq\frac{1}{n^{2}}(1-\varepsilon).
\]

\end{theorem}

\textbf{Acknowledgements.} This research was supported in part by the
NSF\ grant DMS--1811373. The first author would like to offer special thanks
to the University of Rouen Normandie, where a large portion of this research
was accomplished during her visit of the Department of Mathematics. The
authors wish to thank the referee for many in-depth comments, suggestions, and
corrections, which have greatly improved the manuscript.


text:
WWE 1CD 20510
Alfred Schnittke
orchestral works & chamber music
Prelude in memoriam 04:31
Concerto Grosso No. 1 27:24
Two Short Pieces 07:33
Trio-Sonata 29:15
Alfred Schnittke orchestral works & chamber music 09:16
Music for koto over the centuries: from the forefather of Japanese music Yatsuhashi kengyô (1614-1685) to its great-grandson Toshio Hosokawa (*1955).
The documentation of the Donaueschinger Musiktage 2000 presented with pride
by col legno!
"The succinctness of his work will first become fully apparent when it becomes possible to view the second half of the twentieth century from something more like a bird's-eye view." (Alessandro Solbiati)

redpajama_set_name: RedPajamaCommonCrawl


Volunteers from Angus and Dundee bagged a successful beach clean at the weekend, removing at least a ton of rubbish from the Barry Buddon and Monifieth area.
Twenty bags of rubbish were cleared from a north-east beach on Sunday in memory of a kayaker who drowned last year.
Almost 9,000 items of rubbish, including 2,000 cotton buds and even a safety cone, have been removed from Fife beaches this year.
The Dundee branch of the Marine Conservation Society has raised concerns over the amount of dog faeces and rubbish picked up during a beach clean.


Every 5th unexcused tardy will be counted, for determination of school truancy, as an unexcused absence. Students who are late to school in the morning and/or who are picked up from school early will be tardy for the day and will be either excused or unexcused based on Board policy.
The parent must provide written documentation within 3-days detailing the specific reason that the student is late to school or is being taken out of school before dismissal. The failure to provide written documentation within 3-days will result in an unexcused tardy.
If the student is either late to school or being picked up early due to a medical appointment, it is recommended that the parent provide the school with evidence of the appointment. In such cases, the tardy will be excused.
School Board policies on attendance can be found in the Student Progression Plan on the district website at www.stlucieschools.org. Click on "Parents" and look for "Student Progression" on the left. Students with 5 unexcused absences in a 30-day period or 10 in a 90-day period will likely have a Truancy Petition filed in Circuit Court.
Students who are late to school and/or picked up early miss valuable instructional time. This also causes a disruption to the classroom and impacts the learning of all students in the class. It is recognized that there may be occasional times when being late or needing to leave early is unavoidable. If the reason for being late or leaving early is included as "excused" in the Student Progression Plan, the tardy will be excused.
Please contact the attendance office at the school if you have any questions.


Okanagan Lake provides a beautiful setting for the city of Kelowna, one of the fastest growing cities on the continent. Located in southern British Columbia in western Canada, the Okanagan Valley climate is perfect for growing grapes, and more than a hundred vineyards are located in the region. The city boasts a metro population of about 180,000 and is served by Kelowna International Airport (YLW). City bus is operated by the Kelowna Regional Transit System.
Kelowna is home to the University of British Columbia Okanagan; if you plan to attend a Heat athletic event, choose a hotel near UBC Okanagan to be near the game and other campus activities. You might also enjoy a hotel near City Park which is a popular location for Okanagan Lake beach and water-related activities. The annual Center of Gravity festival is celebrated at City Park in August, with 3 stages of live music, beach volleyball, wakeboarding, skateboarding and freestyle mountain biking.
Year-round outdoor recreation reigns supreme in Kelowna. Two major ski destinations are Big White Ski Resort and Silver Star Mountain, and you can enjoy meticulous ski trails, outstanding terrain parks, sleigh rides, snowmobiling, ice skating and other active pursuits. Okanagan Lake Provincial Park is only 30 minutes from Kelowna; this park also offers scenic mountain views, and visitors flock here to spy local birds and other wildlife and to canoe, kayak, fish, boat, hike and bicycle. Try hiking, biking or horseback riding the Myra Canyon Trestle Trail in Myra-Bellevue Provincial Park for outstanding canyon vistas and to see wildlife. Yet another popular park, Fintry Provincial Park, has preserved a region boasting mountains, canyons, a lake with beaches, and even an amazing triple waterfall!
Another fun way to explore the beautiful Okanagan Valley is by touring some of its local vineyards. Mission Hill Winery is one of the favorites, offering tours with tastings and a much-loved Summer Concert Series. Tour Quails' Gate Winery and have a gourmet meal at its Old Vines Restaurant, which boasts lovely Lake Okanagan views. Each season the region celebrates an Okanagan Wine Festival, so grab a few friends and partake in tasting various vineyards' favorite wines, delicious foods and special entertainment.
It's clear that Kelowna is a wonderful destination for outdoor pursuits. But is there anything cultural here? Glad you asked! The Kelowna Art Gallery boasts a nice collection of contemporary Canadian Artwork. The city's performing arts centre, the Rotary Centre for the Arts, offers a variety of productions and hands-on workshops and activities as well. An annual ArtWalk featuring hundreds of artists' work and live musical entertainment is offered at Lake Country Community Complex each September. Great concerts and large events are held at Prospera Place, which is also the home of the WHL Kelowna Rockets. Prospera Place hosts a fun Canada Day celebration, offering an arts and crafts marketplace, food fair, kids' zone, concerts and fireworks. Additionally, the Okanagan Symphony Orchestra and Kelowna Ballet perform at the Kelowna Community Theatre. The arts are alive and well in Kelowna!
Still looking for more reasons to visit Kelowna? How about a traditional Japanese garden, Kasugai Gardens, or the Okanagan Lavender and Herb Farm, where you can explore a real working farm – especially fun during the summer's Lavender Discovery Days festival. Golf is popular in this region, with The Harvest Golf Club being one of the most challenging courses around. Orchard Park Shopping Centre lets you shop til you drop, and it's always fun to browse the Kelowna Farms' and Crafters' Market which is open year-round.
Kelowna is not only visually appealing, there's more to do and see than you ever imagined. We have gathered together the details about all the best Kelowna attractions and events for you. All you have to do is make your reservations!


Back in the old days, the leading browser for Mac OS X was Internet Explorer. That's right, kids — Microsoft used to write Macintosh software.
In addition to offering better web standards support than any other browser in that era, IE for Mac was also quite scriptable. You could, in fact, make minor adjustments to our Save and Restore Safari URLs scripts posted earlier today, and use them with IE.
And, remarkably, aftering making these changes, the same scripts could be used both on Mac OS X and on the now-long-forgotten Mac OS 9 (a relic so old that it used the rainbow-colored Apple logo; ask your grandpa about it).
set prefs_file to prefs_folder & "Internet Explorer Saved URLs"
Combining Safari, AppleScript and the nature of the OS X system, a new and more efficient way of browsing can be created that makes the question of tabbed browsing somewhat irrelevant.


text:
'Profiles in Corruption'
Schiff Desperate
'Thanks Famous Actors': Voters Hit Back at Pro-Clinton Celebrity PSAs
Daniel Nussbaum
Celebrities, and particularly movie stars, have an irritating tendency to believe that just because they were in That Big Movie from ten years ago, the American public should automatically care about their political opinions or which candidate they're going to vote for in this year's election. That's why so many of them are apparently lining up to do not-at-all-subtle anti-Donald Trump "voter registration" public service announcements.
Well, one group of everyday Americans is here to send a message to these so-called celebs: they're absolutely right. How would we know when Election Day is without the guy from Agents of S.H.I.E.L.D. reminding us?
That's the gag behind a mock PSA from "adversarial advertising" company Brabender Cox, which gathered a group of regular (not-famous) Americans together to see what they think about the deluge of celebrity voting reminders.
"Election Day is Tuesday, November 8th. I know this because some famous actors told me," the video's participants deadpan to the camera. "Thanks famous actors. I had no idea."
"They know stuff we just don't know. Stuff we can't know. Because we're not famous actors. And they're not just acting smart. They are smarter," the group adds.
Based on its title, the video is a direct response to one of the more well-publicized celebrity voter PSAs released recently; Joss Whedon's "Important – Save the Day" spot, in which he gathered together his cast from The Avengers — Robert Downey Jr., Scarlett Johansson, Don Cheadle and Mark Ruffalo — and a handful of other stars to remind Americans that it is, like, super important to vote in this year's election, you guys.
"You might think it's not important, you might you're not important," Mark Ruffalo tells us in that clip, released last month. "But that's not true."
"And the only way we can prove that to you is by having lots and lots of famous people repeating how important it is," adds Downey Jr.
In fact, the only way they could get all those famous people together is because of the heightened importance of this year's election. Take it from Don Cheadle: we could be stuck with a "racist, abusive coward who could permanently damage the fabric of our society."
Well, thanks Hollywood actors, for reminding us not to vote for Trump.
"I could have made a terrible mistake without their help," one (not famous) woman says in the first clip above. "I thought [Hillary] Clinton deleting all those emails was dishonest and corrupt, and put our national security at risk. But now we know it was all just an accident! An accident that could happen to any of us. Who among us couldn't accidentally hit the delete button – thirty thousand times?"
Follow Daniel Nussbaum on Twitter: @dznussbaum
EntertainmentPoliticsDonald TrumpHillary ClintonJoss WhedonMark RuffaloRobert Downey Jr.


There has been a great deal of anecdotal evidence to suggest that weather affects consumer decision making. In this paper, we provide empirical evidence to explain how the weather affects consumer spending and we detail the psychological mechanism that underlies this phenomenon. Specifically, we propose that the effect of weather – and, in particular, sunlight – on consumer spending is mediated by negative affect. That is, as exposure to sunlight increases, negative affect decreases and consumer spending tends to increase. We find strong support for this prediction across a series of three mixed methods studies in both the lab and the field.


A series of film and video recommendations.
This web series documents Black German biographies in contemporary Germany.
Polyglot is a web series following the life of an Afro-German woman in Berlin.
Witnessed is a series of short clips featuring black authors and artists in Germany.
Based in Switzerland and Ghana, a series of discussions on Black literature as a tool for critical knowledge, sharing, and societal and personal change.
Film footage of Nazi Germans shouting gibes at French colonial troops.
A clip from the NDR documentary, Ein Deutscher Boxer (2012) .
A personal account of the racist comments a black German ecounters in his everyday life.
A comedic treatment of racial comments encountered in everyday life.
The granddaughter of a Nazi war criminal on Deutsche Welle.
This 2016 documentary uses interviews with figures like Sami Deluxe, Gerald Asamoah, and Theodor Michale to explore what it means to be Black in Germany today.
This 2014 documentary follows the life and career of Black actor Majub bin Adam Mohamed Hussein, who eventually was deported to the concentration camp Sachsenhausen, where he died in 1944.
A documentary about Afro-German youth.
A documentary on the history of German hip hop, featuring several Afro-German musicians.
Toxi tells the story of a young Afro-German girl of the same name who finds herself living with a white middle-class German family.
Pepe Danquart's Oscar-winning short film from 1993 captures the daily racism a black man experiences on the tram.
Teza tells the story of an Ethiopian student migrating back and forth between Ethiopia, West Germany, and East Germany.
Kleine Große Stimme tells the tale of an Austrian boy seeking his African American father.
Premiering in 2006 on German television, it is also available with English subtitles.


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I think this is one of my most unique patterns to date. The beautiful blues and greens will make a striking impact on a wall!
Finished piece will be about 19x13 inches, based on a 14 ct. aida cloth (although you can use any size/brand you like).


The United States is home to some 67,000 pharmacies. Every year, these facilities dispense about 4.4 billion drug prescriptions.
No wonder the U.S. pharmaceuticals market has become a $934.8 billion industry! What's more, experts estimate that come 2021, this industry will be worth $1.17 trillion.
But so will the prices of the drugs they sell and dispense. The U.S., after all, is the only Western country that doesn't have drug pricing regulations. That's why prescription drugs here are 50% to 80% more expensive than in Canada.
This huge price difference is why many turn to legitimate Canadian pharmacies online. These e-pharmacies not only offer big discounts, but also convenience and privacy. But first, you need to know how to spot the legit ones.
We'll tell you exactly how to do that, so keep reading!
When ordering prescriptions from Canada, do so through a retail online pharmacy. These are pharmacies that have both brick and mortar stores and an online drugstore. That means patients can actually go to a physical pharmacy as a walk-in patient!
Having a retail address may give you more peace of mind since you know the pharmacy's exact location. Patients know where to find them, and so do the authorities.
This doesn't mean that online pharmacies without retail stores are fakes. But those with a physical store may be your better, safer choice.
Buy your prescriptions only from a Canadian International Pharmacy Association (CIPA) member. A CIPA certification means the government has licensed and regulated the pharmacy. Their products have passed product safety testing.
Also, look for licensure from other licensing bodies, like Canada's College of Pharmacists. Another is approval from the Internet Mail-Order Pharmacy Accreditation Commission (IMPAC).
Check as well for a Verified Internet Pharmacy Practice Sites (VIPPS) seal. This seal means that the pharmacy is a legitimate business.
Say you're one of the 40 million U.S. adults suffering from anxiety disorders. You're taking Xanax now but would like to know your other options, especially generics.
Of course, your best (and first) source of information is still your own doctor. But you can also use this is a way to gauge the legitimacy of an online pharmacy. When you call the pharmacy, ask if you can speak to a pharmacist for questions about your medication.
A legitimate pharmacy will let you speak to one right away. Having a licensed pharmacist on board is one of the signs they're a legit business. In fact, they'll recommend talking to a pharmacist first if you plan to make changes to your meds.
If you encounter a website that won't let you speak to a pharmacist, that's a red flag. Stay away from such websites as they're most likely fake businesses. They're supposed to be a pharmacy, so a pharmacist should always be available!
All legit pharmacies require patients to first show their prescription. Even if you're buying prescriptions from Canada. Unless you send over a copy of it, they won't process your orders.
Don't worry though, as you have several ways to send your prescription over. The easiest is to upload your prescription through the pharmacy's website. Most reputable online drugstores have an Rx uploading tool built-in on their site.
They also have specific email addresses where you can send a scanned copy of your Rx. Faxing your prescription is also a legal and quick way to show them this document. You also have the snail mail option, although that'll take a long time.
But what if you need to order affordable diabetes drugs now but you don't have your Rx with you? Then make sure you leave your doctor's contact information with the online pharmacy. The pharmacy's staff will call your doctor to request your prescription.
Whereas rogue pharmacies will sell you drugs even without seeing your prescription. They won't ask you to send them a copy of this document and may even tell you that you don't have to. They may even offer to write you a prescription.
Selling prescription drugs without a prescription is illegal, both in the U.S. and in Canada. Also, only licensed telemedicine doctors can write prescriptions online. A website telling you otherwise is likely a fake or rogue pharmacy.
Reputable online pharmacies make use of the latest in digital encryption and security. For instance, there's GeoTrust SSL, which is one of today's best encryption technology. This provides one of the highest levels of security to websites.
If you can't find any logo or information about the site's security measures, stay away from it. It may leak or even expose your sensitive information to hackers.
Aside from the website, look for other platforms where you can find the pharmacy. Google My Business should be one of them, as it displays the exact location, contact info, and even reviews. You'll also see photos of retail pharmacies uploaded by actual consumers.
You should also check their ratings and reviews on Facebook if they're on this social media site. Read as many of the reviews as you can, so you can have a better idea on the pharmacy. You'll learn about their prices, quality of customer service, or if they're even a legit business at all.
Then it most likely is. We mentioned that Canadian prescription drugs can be up to 80% cheaper than U.S. drugs. But if it's even cheaper than that, then that may already be a sign you're dealing with a fake business.
Don't buy from these websites because they may be cybercriminals. They'll only steal your identity and financial information.
Follow these tips, and you'll have no problems finding legitimate Canadian pharmacies online. So long as you know what to look for, you won't fall victim to fake business and rogue pharmacies. Also, if you do find a scam website, be sure to report it!
Want even more tips to make your prescription drugs even more affordable? Then check out our guide on the best money-saving hacks when buying prescriptions!
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With 1200+ happy clients and 20+ years experience, Imaginet can help you with your Microsoft SharePoint deployment, setup, and configuration so that you get your SharePoint right... the first time.
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Most organizations lack the experience to know how to successfully deploy and configure Microsoft SharePoint, from start to finish. This is where Imaginet can help. With 1200+ happy clients and 20+ years of experience, Imaginet is your trusted, go-to partner for all of your SharePoint Deployment Services needs.
Imaginet's SharePoint Deployment Services Quick Start is designed to assist organizations with the installation, setup, and configuration of Microsoft SharePoint with minimum downtime and maximum ROI. Our SharePoint Deployment Services Quick Start can be broken down into three distinct phases: Plan, Deploy, and Review.
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As part of our SharePoint Deployment Services Quick Start, Imaginet can help your team not only understand the core SharePoint capabilities, but also the new SharePoint features available in the version you're deploying.
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