Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 26, 2007, 17–25

© K. Budsaba, P. Chen, A. Volodin

Abstract. Let $\left\{Y_i,-\infty <i<\infty \right\}$ be a doubly infinite sequence of identically distributed ${\rho }^{-}$-mixing or negatively associated random variables, $\left\{a_i,-\infty <i<\infty \right\}$ a sequence of real numbers. In this paper, we prove the rate of convergence and strong law of large numbers for the partial sums of moving average processes $\left\{{\sum }_{i=-\infty }^{\infty }{a}_i{Y}_{i+n},n\ge 1\right\}$ under some moment conditions.

1. Preliminaries

Let $\left\{Y_i,-\infty <i<+\infty \right\}$ be a doubly infinite sequence of identically distributed random variables and $\left\{a_i,-\infty <i<+\infty \right\}$ be an absolutely summable sequence of real numbers. Next, let

be the moving average process based on the sequence $\left\{Y_i,-\infty <i<+\infty \right\}$. As usual, we denote $S_n={\sum }_{k=1}^n{X}_k,n\ge 1$, the sequence of partial sums.

Under the assumption that $\left\{Y_i,-\infty <i<+\infty \right\}$ is a sequence of independent identically distributed random variables, many limiting results have been obtained for the moving average process $\left\{X_n,n\ge 1\right\}$. For example, Ibragimov (1962) established the central limit theorem, Burton and Dehling (1990) obtained a large deviation principle, and Li, Rao, and Wang (1992) obtained the complete convergence result for $\left\{X_n,n\ge 1\right\}$.

Certainly, even if $\left\{Y_i,-\infty <i<+\infty \right\}$ is the sequence of independent identically distributed random variables, the moving average random variables $\left\{X_n,n\ge 1\right\}$ are dependent. This kind of dependence is called weak dependence. The partial sums of weakly dependent random variables $\left\{X_n,n\ge 1\right\}$ have similar limiting behaviour properties in comparison with the limiting properties of independent identically distributed random variables.

Very few results for a moving average process based on a dependent sequence are known. In this paper, we provide two results on the limiting behaviour of a moving average process based on a ${\rho }^{-}$-mixing and negatively associated sequences.

Let $\left\{Y_i,-\infty <i<\infty \right\}$ be a sequence of random variables defined on a probability space $\left(\Omega ,\mathcal{ℱ},P\right)$. For a set of integer numbers $T$ denote $\sigma$-algebra $\mathcal{ℱ}\left(T\right)=\sigma \left(Y_i,i\in T\right)$ and as usual, for a $\sigma$-algebra $\mathcal{ℱ}$ we denote by ${\mathcal{ℒ}}^2\left(\mathcal{ℱ}\right)$ the class of all $\mathcal{ℱ}$-measurable random variables with the finite second moment.

For two sets $S$ and $T$ of real numbers we denote

The following definition was introduced in Zang and Wang (1999). A sequence of random variables $\left\{Y_i,-\infty <i<\infty \right\}$ is called ${\rho }^{-}$-mixing if

as $s\to \infty$, where

where supremum is taken over all coordinatewise increasing real functions $f$ on $R^S$ and $g$ on $R^T$ and by $Corr\left(\cdot ,\cdot \right)$ we denote the classical correlation coefficient.

Next, a sequence of random variables $\left\{Y_i,-\infty <i<\infty \right\}$ is called ${\rho }^{\ast }$-mixing if for some integer $s\ge 1$

where the first sup is taken over all pairs of nonempty finite sets $S,T$ of integers, such that dist$\left(S,T\right)\ge s$. The notion of ${\rho }^{\ast }$-mixing seems to be similar to the notion of $\rho$-mixing, but Bryc and Smolenski (1993) showed that they are quite different from each other.

Recall that a finite family of random variables $\left\{Y_i,1\le i\le n\right\}$ is said to be negatively associated, if for any disjoint finite subsets $S$ and $T$ of integers and any real coordinatewise nondecreasing functions $f$ on $R^S$ and $g$ on $R^T$,

whenever the covariance exists. This concept was studied in Joag-Dev and Proschan (1983).

It is easy to see that $\left\{Y_i,-\infty <i<\infty \right\}$ is negatively associated if and only if ${\rho }^{-}\left(s\right)=0$ for all $s\ge 1$ and ${\rho }^{-}\left(s\right)\le {\rho }^{\ast }\left(s\right)$. Hence the notion of ${\rho }^{-}$-mixing is weaker than both notions of negative association and ${\rho }^{\ast }$-mixing.

We also need the following simple statement (see Property P2 in Wang and Lu (2006))

Property of ${\rho }^{-}$-mixing random variables. Let $\left\{Y_n,n\ge 1\right\}$ be a sequence of ${\rho }^{-}$-mixing random variables. If $\left\{f_n,n\ge 1\right\}$ is a sequence of real functions all of which are monotone nondecreasing (or all monotone nonincreasing), then $\left\{f_n\left(Y_n\right),n\ge 1\right\}$ is a sequence of ${\rho }^{-}$-mixing random variables.

Note that Property P2 in Wang and Lu (2006) is stated only for increasing functions. It is simple to see that this property remains true for nondecreasing functions, too. The statement for nonincreasing functions follows for the observation that if a function $f_n$ is nonincreasing, then the function $-f_n$ is nondecreasing.

Recall that a measurable function $l$ is said to be slowly varying if for each $\lambda >0$

We refer to Seneta (1976) for other equivalent definitions and for detailed and comprehensive study of properties of such functions.

In the following, $C$ will represent a positive constants although its value may change from one appearance to the next.

The following result was proved in Budsaba, Chen, and Volodin (2007).

Theorem BCV. Let $l\left(x\right)$ be a positive slowly varying function and $1\le p<2,r\ge 1,pr\ne 1$. Suppose $\left\{Y_i,-\infty <i<\infty \right\}$ is a sequence of identically distributed and ${\rho }^{-}$-mixing random variables and $\left\{X_n,n\ge 1\right\}$ is the moving average process based on the sequence $\left\{Y_i,-\infty <i<\infty \right\}$. Then $E{Y}_1=0$ and $E\mid Y_1{\mid }^{rp}l\left(\mid Y_1{\mid }^p\right)<\infty$ imply that for all $\varepsilon >0$

In particular, the assumptions $E{Y}_1=0$ and $E\mid Y_1{\mid }^p<\infty ,1<p<2$ imply Marcinkiewicz-Zygmund strong law of large numbers

Next, in Concluding Remark 4, Budsaba, Chen, and Volodin (2007) mentioned that the case $p=r=1$ is not treated in Theorem BCV and that the authors believe that the result can be proved under the additional assumption ${\sum }_{i=-\infty }^{\infty }\mid a_i{\mid }^s<\infty$ for some $0<s<1$.

In this paper it is shown that this suggestion is true. Namely, we prove the following result.

Theorem 1. Let $l\left(x\right)$ be a positive slowly varying function and suppose that $\left\{Y_i,-\infty <i<\infty \right\}$ is a sequence of identically distributed ${\rho }^{-}$-mixing random variables. Let $\left\{X_n,n\ge 1\right\}$ be the moving average process based on the sequence $\left\{Y_i,-\infty <i<\infty \right\}$. Let moreover $\left\{a_i,-\infty <i<\infty \right\}$ be a sequence of real numbers with ${\sum }_{i=-\infty }^{\infty }\mid a_i{\mid }^s<\infty$ for some $0<s<1$. Then $E{Y}_1=0$ and $E\mid Y_1\mid l\left(\mid Y_1\mid \right)<\infty$ imply that for all $\varepsilon >0$

In particular, $E{Y}_1=0$ implies the following Kolmogorov strong law of large numbers

Proof. Let $Y_{nj}^{\left(1\right)}=-nI\left(Y_j<-n\right)+Y_{i}I\left(\mid Y_j\mid \le n\right)+nI\left(Y_j>n\right)$, and $Y_{nj}^{\left(2\right)}=Y_j-Y_{nj}^{\left(1\right)}$. Then by Property of ${\rho }^{-}$-mixing random variables, for any $n\ge 1$, $\left\{Y_{nj}^{\left(1\right)},-\infty <j<\infty \right\}$ and $\left\{Y_{nj}^{\left(2\right)},-\infty <j<\infty \right\}$ are sequences of ${\rho }^{-}$-mixing random variables. Note that

and

as $n\to \infty$. Hence for $n$ large enough

Therefore it is enough to prove that

and

For $I_2$, by Markov inequality, we have

For $I_1$, by Markov and Hölder inequalities, we have

Now we show the almost sure convergence. By the first part of Theorem, $E{Y}_1=0$ (and hence $E\mid Y_1\mid <\infty$) implies

Therefore

By Borel-Cantelli lemma

which implies that $S_n∕n\to 0\text{ almost surely.}\square$

The second theorem treats the case when the sequence $\left\{a_i,-\infty <i<+\infty \right\}$ is not absolutely summable.

Theorem 2. Let $1<q\le 2$. Assume that there exists $s,1<s<q$, such that

Suppose $\left\{Y_i,-\infty <i<\infty \right\}$ is a sequence of identically distributed negatively associated random variables and $\left\{X_n,n\ge 1\right\}$ is the moving average process based on the sequence $\left\{Y_i,-\infty <i<\infty \right\}$. Then for all $p,0<p<\frac{sq}{sq-q+s}$, the assumptions $E{Y}_1=0$ and $E\mid Y_1{\mid }^q<\infty$ imply Marcinkiewicz-Zygmund strong law of large numbers

Remark. Note that the assumption $1<s<q\le 2$ implies that $p<2$.

Proof. Without loss of generality, we assume that $a_i\ge 0$ for all $i$. Let $\left\{b_n,n\ge 1\right\}$ be sequence of positive numbers that will be specified later. By Theorem 2 (1.6) of Shao (2000), we have

An application of Hölder and then Jensen inequalities yields

Hence

By Theorem 3.3 of Móricz, Serfling, and Stout (1982), we have the following maximal inequality

This maximal inequality implies the almost sure convergence of the series ${\sum }_{k=1}^{\infty }{b}_k{X}_k$ as soon as ${\sum }_{k=1}^{\infty }{b}_k^q{k}^{q\left(1-1∕s\right)}{log}_2^q\left(2k\right)<\infty$ (see for instance, Loève (1978) Section 36.1). The application of Kronecker lemma with $b_k=k^{-1∕p}$ concludes the proof of Theorem 2. $\square$

References

Department of Mathematics and Statistics, Thammasat University, Rangsit Center, Pathumthani, 12121, Thailand

E-mail address: kamon@mathstat.sci.tu.ac.th

Department of Mathematics, Jinan University, Guangzhou, 510630, China

E-mail address: chenpingyan@yahoo.com.cn

Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada, S4S 0A2

E-mail address: andrei@math.uregina.ca

Received March 18, 2007