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# On almost sure limiting behavior of a dependent random sequence

Journal of Inequalities and Applications20132013:25

https://doi.org/10.1186/1029-242X-2013-25

• Received: 28 August 2012
• Accepted: 3 January 2013
• Published:

## Abstract

We study some sufficient conditions for the almost certain convergence of averages of arbitrarily dependent random variables by certain summability methods. As corollaries, we generalized some known results.

MSC:60F15.

## Keywords

• dependent random variable
• summability
• dominated random sequence

## 1 Introduction

In reference , Chow and Teicher gave a limit theorem of almost certain summability of i.i.d. random variables as follows.

Theorem (Chow et al., 1971)

Let $a\left(x\right)$, $x>0$ be a positive non-increasing function and ${a}_{n}=a\left(n\right)$, ${A}_{n}={\sum }_{k=1}^{n}{a}_{k}$, ${b}_{n}={A}_{n}/{a}_{n}$, where
1. (1)

${A}_{n}\to \mathrm{\infty }$;

2. (2)

$0<{lim inf}_{n}\frac{{b}_{n}}{n}a\left(log{b}_{n}\right)\le {lim sup}_{n}\frac{{b}_{n}}{n}a\left(log{b}_{n}\right)<\mathrm{\infty }$;

3. (3)
$xa\left({log}^{+}x\right)$ is non-decreasing for $x>0$, then i.i.d. $\left\{X,{X}_{n}\right\}$ are ${a}_{n}$ summable, i.e.,
${T}_{n}={A}_{n}^{-1}\sum _{k=1}^{n}{a}_{k}{X}_{k}-{C}_{n}\to 0\phantom{\rule{1em}{0ex}}\mathit{\text{a.c.}}$

for some choice of centering constants ${C}_{n}$, if and only if
$E|X|a\left({log}^{+}|X|\right)<\mathrm{\infty }.$

Motivated by Chow and Teicher’s idea, in this paper we consider the problem of arbitrarily dependent random variables and their limiting behavior from a new perspective.

Throughout this paper, let denote the set of positive integers, $\left\{X,{X}_{n},{\mathcal{F}}_{n},n\in \mathbb{N}\right\}$ be a stochastic sequence defined on the probability space $\left(\mathrm{\Omega },\mathcal{F},\mathbb{P}\right)$, i.e., the sequence of σ-fields $\left\{{\mathcal{F}}_{n},n\in \mathbb{N}\right\}$ in is increasing in n, and $\left\{{\mathcal{F}}_{n}\right\}$ are adapted to random variables $\left\{{X}_{n}\right\}$, ${\mathcal{F}}_{0}$ denotes the trivial σ field $\left\{\mathrm{\Phi },\mathrm{\Omega }\right\}$ and ${\mathbf{1}}_{\left[\cdot \right]}$ the indicator function.

We begin by introducing some terminology and lemmas.

Definition 1 (Adler et al., 1987 )

Let $\left\{{X}_{n},n\in \mathbb{N}\right\}$ be a sequence of random variables, and it is said to be stochastically dominated by a random variable X (we write $\left\{{X}_{n},n\in \mathbb{N}\right\}\prec X$) if there exists a constant $C>0$, for almost every $\omega \in \mathrm{\Omega }$, such that

Lemma 1 (Chow et al., 1978 )

Let $\left\{{X}_{n},{\mathcal{F}}_{n},n\in \mathbb{N}\right\}$ be an ${L}_{p}$ ($1\le p\le 2$) martingale difference sequence, if ${\sum }_{n=1}^{\mathrm{\infty }}E\left({|{X}_{n}|}^{p}|{\mathcal{F}}_{n-1}\right)<\mathrm{\infty }$, then ${\sum }_{n=1}^{\mathrm{\infty }}{X}_{n}$ a.c. converges.

Lemma 2 Let $\left\{X,{X}_{n},n\in \mathbb{N}\right\}$ be a sequence of random variables. If $\left\{{X}_{n}\right\}\prec X$, then for all $t>0$,
$\mathbb{E}{|{X}_{n}|}^{2}{\mathbf{1}}_{\left[|{X}_{n}|\le t\right]}\le C\left[{t}^{2}\mathbb{P}\left(|X|>t\right)+\mathbb{E}{X}^{2}{\mathbf{1}}_{\left[|X|\le t\right]}\right].$
Proof By the integral equality
$2{\int }_{0}^{t}s\mathbb{P}\left(|{X}_{n}|>s\right)\phantom{\rule{0.2em}{0ex}}ds={t}^{2}\mathbb{P}\left(|{X}_{n}|>t\right)+\mathbb{E}{|{X}_{n}|}^{2}{\mathbf{1}}_{\left[|{X}_{n}|\le t\right]},$

□

## 2 Strong law of large numbers

In this section, we always assume that $a\left(x\right)$, $x>0$ is a positive non-increasing function and ${a}_{n}=a\left(n\right)$, ${A}_{n}={\sum }_{k=1}^{n}{a}_{k}$, ${b}_{n}={A}_{n}/{a}_{n}$, where
1. (1)

${A}_{n}\to \mathrm{\infty }$;

2. (2)

$0<{lim inf}_{n}\frac{{b}_{n}}{n}a\left(log{b}_{n}\right)\le {lim sup}_{n}\frac{{b}_{n}}{n}a\left(log{b}_{n}\right)<\mathrm{\infty }$;

3. (3)

$xa\left({log}^{+}x\right)$ is non-decreasing for $x>0$.

Theorem 1 Let $\left\{X,{X}_{n}\right\}$ be a sequence of random variables with $\left\{{X}_{n}\right\}\prec X$. If $E|X|a\left({log}^{+}|X|\right)<\mathrm{\infty }$, then
$\underset{n}{lim}\frac{1}{{A}_{n}}\sum _{k=1}^{n}{a}_{k}\left[{X}_{k}-E\left({X}_{k}{\mathbf{1}}_{\left[|{X}_{k}|\le {b}_{k}\right]}|{\mathcal{F}}_{k-1}\right)\right]=0,\phantom{\rule{1em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathit{\text{a.c.}}$
(2.1)
Proof To prove (2.1) by applying the Kronecker lemma, it suffices to show that
Since $0, (1) guarantees that ${b}_{n}↑\mathrm{\infty }$. Choose ${m}_{0}$ such that $n\ge {m}_{0}$ implies
$\alpha n\le {b}_{n}a\left(log{b}_{n}\right)\le \beta n$
(2.2)
whence ${b}_{n}\ge \alpha n{\left[a\left(log{b}_{m}\right)\right]}^{-1}$ for $n\ge m\ge {m}_{0}$ entailing
$\sum _{k=m}^{\mathrm{\infty }}{b}_{k}^{-2}\le \frac{{a}^{2}\left(log{b}_{m}\right)}{{\alpha }^{2}m}.$
(2.3)
Put ${Y}_{n}={X}_{n}{\mathbf{1}}_{\left[|{X}_{n}|\le {b}_{n}\right]},{Z}_{n}={X}_{n}{\mathbf{1}}_{\left[|{X}_{n}|>{b}_{n}\right]}$, obviously, ${X}_{n}={Y}_{n}+{Z}_{n},n\in \mathbb{N}$. Note that $\left\{{X}_{n}\right\}\prec X$ and the condition $E|X|a\left({log}^{+}|X|\right)<\mathrm{\infty }$, we have
which shows
$\mathbb{P}\left({X}_{n}\ne {Z}_{n},\text{i.o.}\right)=0.$
(2.5)

Let ${W}_{n}=\frac{{Y}_{n}}{{b}_{n}}-E\left(\frac{{Y}_{n}}{{b}_{n}}|{\mathcal{F}}_{n-1}\right)$, then $\left({W}_{n},{\mathcal{F}}_{n},n\in \mathbb{N}\right)$ is a martingale difference sequence.

Note that
$\begin{array}{rcl}E\left[\sum _{n=1}^{\mathrm{\infty }}E\left({W}_{n}^{2}|{\mathcal{F}}_{n-1}\right)\right]& \le & E\left[\sum _{n=1}^{\mathrm{\infty }}E\left(\frac{{Y}_{n}^{2}}{{b}_{n}^{2}}|{\mathcal{F}}_{n-1}\right)\right]\\ =& \sum _{n=1}^{\mathrm{\infty }}E\frac{{Y}_{n}^{2}}{{b}_{n}^{2}}<\mathrm{\infty },\end{array}$
(2.7)

which implies that ${\sum }_{n=1}^{\mathrm{\infty }}E\left({W}_{n}^{2}|{\mathcal{F}}_{n-1}\right)<\mathrm{\infty }$ a.c. Hence, by Lemma 1, we have ${\sum }_{n=1}^{\mathrm{\infty }}{W}_{n}$ a.c. convergence.

Theorem 1 follows from (2.5) and (2.7). □

Theorem 1 also includes some particular cases of means, we can establish the following.

Corollary 1 Let $\left\{X,{X}_{n},n\in \mathbb{N}\right\}$ be a sequence of random variables with $\left\{{X}_{n}\right\}\prec X$. If for some $\epsilon >0$, $E\frac{|X|}{log|X|{\mathbf{1}}_{\left[|X|>\epsilon \right]}}<\mathrm{\infty }$, then
$\underset{n}{lim}\frac{1}{logn}\sum _{k=1}^{n}\left[\frac{{X}_{k}-E\left({X}_{k}{\mathbf{1}}_{\left[|{X}_{k}|\le klogk\right]}|{\mathcal{F}}_{k-1}\right)}{k}\right]=0,\phantom{\rule{1em}{0ex}}\mathit{\text{a.c.}}$
Corollary 2 Let $\left\{X,{X}_{n},n\in \mathbb{N}\right\}$ be a sequence of random variables with $\left\{{X}_{n}\right\}\prec X$ and for some $k\ge 2$,
${a}_{n}={\left[n\left(logn\right)\cdots \left({log}_{k-1}n\right)\right]}^{-1},$
where ${log}_{1}n=logn$, ${log}_{k}n=log\left({log}_{k-1}n\right)$, $k\ge 2$, if for all large $C>0$,
$E\frac{|X|{\mathbf{1}}_{\left[|X|>C\right]}}{\left(log|X|\right)\cdots \left({log}_{k}|X|\right)}<\mathrm{\infty },$
then
$\underset{n}{lim}\frac{1}{{A}_{n}}\sum _{k=1}^{n}{a}_{k}\left[{X}_{k}-E\left({X}_{k}{\mathbf{1}}_{\left[|{X}_{k}|\le {b}_{k}\right]}|{\mathcal{F}}_{k-1}\right)\right]=0,\phantom{\rule{1em}{0ex}}\mathit{\text{a.c.}}$
Corollary 3 Let $\left\{X,{X}_{n},n\in \mathbb{N}\right\}$ be a sequence of random variables with $\left\{{X}_{n}\right\}\prec X$. Further, let ${\mathcal{F}}_{n}=\sigma \left({X}_{1},\dots ,{X}_{n}\right)$ and ${\mathcal{F}}_{-n}=\left\{\varphi ,\mathrm{\Omega }\right\}$, $n\ge 0$. If $E|X|a\left({log}^{+}|X|\right)<\mathrm{\infty }$, then for any $m\ge 1$,
$\underset{n}{lim}\frac{1}{{A}_{n}}\sum _{k=1}^{n}{a}_{k}\left[{X}_{k}-E\left({X}_{k}{\mathbf{1}}_{\left[|{X}_{k}|\le {b}_{k}\right]}|{\mathcal{F}}_{k-m}\right)\right]=0,\phantom{\rule{1em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathit{\text{a.c.}}$
(2.8)
Proof Since $\left\{{X}_{nm+l},{\mathcal{F}}_{nm+l},n\ge 1\right\}$ is an adapted stochastic sequence and $\left\{{X}_{nm+l}\right\}\prec X$, by Theorem 1, we have for $l=0,1,\dots ,m-1$ that
$\sum _{n=1}^{\mathrm{\infty }}\frac{{X}_{nm+l}-E\left({X}_{nm+l}{\mathbf{1}}_{\left[|{X}_{nm+l}|\le {b}_{nm+l}\right]}|{\mathcal{F}}_{\left(n-1\right)m+l}\right)}{{b}_{nm+l}}\phantom{\rule{1em}{0ex}}\text{converges a.c.}$

□

Corollary 4 Let $\left\{{X}_{n},n\in \mathbb{N}\right\}$ be a sequence of m-dependent random variables. Further, let ${\mathcal{F}}_{n}=\sigma \left({X}_{1},\dots ,{X}_{n}\right)$ and ${\mathcal{F}}_{-n}=\left\{\varphi ,\mathrm{\Omega }\right\},n\ge 0$. If there exists a random variable X such that $\left\{{X}_{n}\right\}\prec X$ and $E|X|a\left({log}^{+}|X|\right)<\mathrm{\infty }$, then
$\underset{n}{lim}\frac{1}{{A}_{n}}\sum _{k=1}^{n}{a}_{k}\left[{X}_{k}-E\left({X}_{k}{\mathbf{1}}_{\left[|{X}_{k}|\le {b}_{k}\right]}\right)\right]=0,\phantom{\rule{1em}{0ex}}\mathit{\text{a.c.}}$

Proof Note that $\left\{{X}_{n},n\in \mathbb{N}\right\}$ is a sequence of m-dependent random variables, then $E\left({X}_{n}|{\mathcal{F}}_{n-m}\right)=E{X}_{n}$, Corollary 4 follows directly from Corollary 3. □

Definition 2 (Stout, 1974)

Let $\left\{{X}_{n},n\in \mathbb{N}\right\}$ be a sequence of random variables, and let ${\mathcal{F}}_{n}^{m}=\sigma \left({X}_{n},\dots ,{X}_{m}\right)$. We say that the sequence $\left\{{X}_{n},n\in \mathbb{N}\right\}$ is *-mixing if there exists a positive integer M and a non-decreasing function $\phi \left(n\right)$ defined on integers $n\ge M$ with ${lim}_{n}\phi \left(n\right)=0$ such that for $n>M$, $A\in {\mathcal{F}}_{0}^{m}$ and $B\in {\mathcal{F}}_{m+n}^{\mathrm{\infty }}$, the relation
$|\mathbb{P}\left(A\cap B\right)-\mathbb{P}\left(A\right)P\left(B\right)|\le \phi \left(n\right)\mathbb{P}\left(A\right)\mathbb{P}\left(B\right)$

holds for any integer $m\ge 1$.

It has been proved (cf. ) that the *-mixing condition is equivalent to the condition
$|\mathbb{P}\left(B|{\mathcal{F}}_{0}^{m}\right)-\mathbb{P}\left(B\right)|\le \phi \left(n\right)\mathbb{P}\left(B\right),\phantom{\rule{1em}{0ex}}\text{a.c.}$
for $B\in {\mathcal{F}}_{m+n}^{\mathrm{\infty }}$ and $m\ge 1$ implies
$|E\left({X}_{n+m}|{\mathcal{F}}_{0}^{m}\right)-E{X}_{n+m}|\le \phi \left(n\right)E|{X}_{n+m}|,\phantom{\rule{1em}{0ex}}\text{a.c.}$
(2.9)
Theorem 2 Let $\left\{X,{X}_{n},n\in \mathbb{N}\right\}$ be a sequence of *-mixing random variables with $\left\{{X}_{n}\right\}\prec X$. Further, let ${\mathcal{F}}_{n}=\sigma \left({X}_{1},\dots ,{X}_{n}\right)$ and ${\mathcal{F}}_{-n}=\left\{\varphi ,\mathrm{\Omega }\right\}$, $n\ge 0$. If $max\left\{E|X|,E|X|a\left({log}^{+}|X|\right)\right\}<\mathrm{\infty }$, then
$\underset{n}{lim}\frac{1}{{A}_{n}}\sum _{k=1}^{n}{a}_{k}\left[{X}_{k}-E{X}_{k}{\mathbf{1}}_{\left[|{X}_{k}|\le {b}_{k}\right]}\right]=0,\phantom{\rule{1em}{0ex}}\mathit{\text{a.c.}}$
Proof By Corollary 3, we have, for each $m\ge 1$,
$\underset{n}{lim}\frac{1}{{A}_{n}}\sum _{k=1}^{n}{a}_{k}\left[{X}_{k}-E\left({X}_{k}{\mathbf{1}}_{\left[|{X}_{k}|\le {b}_{k}\right]}|{\mathcal{F}}_{k-m}\right)\right]=0,\phantom{\rule{1em}{0ex}}\text{a.c.}$
Since $\left\{{X}_{n},n\in \mathbb{N}\right\}$ is *-mixing, by (2.8) and (2.9), we obtain

Thus, using the Kroneker lemma, Theorem 2 follows. □

## Declarations

### Acknowledgements

Foundation item: The National Nature Science Foundation of China (No. 11071104), Foundation of Anhui Educational Committee (KJ2012B117) and Graduate Innovation Fund of AnHui University of Technology (D2011025).

## Authors’ Affiliations

(1)
School of Mathematics & Physics, AnHui University of Technology, Ma’anshan, 243002, People’s Republic of China

## References 