Lobachevskii Journal of Mathematics http://ljm.ksu.ru ISSN 1818-9962 Vol. 21, 2006, 45–55

© Pingyan Chen, Tien-Chung Hu, Andrei Volodin

Abstract. We obtain the complete convergence rates for maximums of partial sums of Banach space valued random elements consisting of a moving average process. The corresponding almost sure convergence results for partial sums are derived, too.

1. Introduction

The concept of complete convergence was first introduced by Hsu and Robbin (1947) as follows. A sequence of random variables $\left\{U_n,n\ge \right\}$ is said to converge completely to a constant $c$ if ${\sum }_{n=1}^{\infty }P\left\{\mid U_n-c\mid >\varepsilon \right\}<\infty$ for all $\varepsilon >0$. By the Borel-Cantelli lemma, this implies $U_n\to c$ almost surely (a.s.) and the converse implication is true if the $\left\{U_n,n\ge 1\right\}$ are independent. Hsu and Robbin (1947) proved that the sequence of arithmetic means of independent and identically distributed random variables converges completely to the excepted value if the variance of the summands is finite. Their research was continued by Erdös (1949,1950), and Baum and Katz(1965) among others.

The following generalization of the Hsu and Robbin (1947) result was obtained in Baum and Katz(1965).

Theorem A. Let $\left\{X_n,n\ge 1\right\}$ be a sequence of independent identically distributed random variables, $S_n={\sum }_{k=1}^n{X}_k$, $\beta \ge -1$, and $0<\nu <2$. Then the conditions $E\mid X_1{\mid }^{\left(\beta +2\right)\nu }<\infty$ and $E{X}_1=0$ for the case $\nu \ge 1$, are necessary and sufficient for

It is an interesting problem to investigate the rate of complete convergence for dependent random variables. One of the first results in this direction, that is the rate of complete convergence for moving average sequences was in Li, Rao, and Wang (1992). This gives a partial solution for the sufficiency part of the Baum-Katz statement for $\beta =0$.

Theorem B. Let $\left\{Y_n,-\infty <n<\infty \right\}$ denote a double infinite sequence of independent identically distributed random variables, and let $V_k={\sum }_{i=-\infty }^{\infty }{a}_{i+k}{Y}_i$ for $k\ge 1$ and $S_n={\sum }_{k=1}^n{V}_k$ for $n\ge 1$. If $E{Y}_1=0,E\mid Y_1{\mid }^{2\nu }<\infty ,1\le \nu <2$, then

The question of the rate of convergence of the moving average process for other values of parameter $\beta$ and in Banach space setting was discussed in Ahmed, Giuliano Antonini, and Volodin (2002) and Chen, Sung, and Volodin (2006). In this paper we are interested only in the moving average process taking values in Banach space of Rademacher type (technical definitions will be discussed in the next section), and hence we will present only the following result. It contains the case $\beta >-1$ from Corollary 4.2 of Ahmed, Giuliano Antonini, and Volodin (2002) and the special case $\beta =-1$ from Corollary 3 of Chen, Sung, and Volodin (2006).

Theorem C. Assume that $\left\{Y_i,-\infty <i<\infty \right\}$ is a doubly infinite sequence of independent mean zero random elements taking values in a separable real Rademacher type $p,1<p<2,$ Banach space and is stochastically dominated by a random variable $X$. Let $\left\{a_i,-\infty <i<\infty \right\}$ be an absolutely summable sequence of real numbers and set $V_i={\sum }_{k=-\infty }^{\infty }{a}_{i+k}{Y}_k,i\ge 1$ and $S_n={\sum }_{k=1}^n{V}_k$ for $n\ge 1$. If $E\mid X{\mid }^{\left(\beta +2\right)\nu }<\infty$ where $\beta \ge -1$ and $1\le \nu <p$, then

We should mention that the proofs of the cases $\beta >-1$ from Corollary 4.2 of Ahmed, Giuliano Antonini, and Volodin (2002) and the special case $\beta =-1$ from Corollary 3 of Chen, Sung, and Volodin (2006) are completely different. The initial goal of the present investigation was to find unified proof of Theorem C as it is stated, but it appears that a stronger result can be obtained. Namely, in this paper we consider the rate of complete convergence of maximums of partial sums for moving average process.

The plan of the paper is as follows. In Section 2, we recall some well known definitions relevant to the current work. In Section 3, we prove Theorem D which presents a sufficient condition for the rate of complete convergence of maximums of partial sums for moving average process. As in Theorems B and C, Theorem D contains an assumption concerning the geometry of the underlying Banach space, namely it is assumed that it is of the Rademacher type $p$. In Section 4, we present a necessary and sufficient result for almost sure convergence of the moving average process. Finally, in Section 5, we provide an additional result for the rate of complete convergence of supremums of normed partial sums for moving average process.

2. Preliminaries

Let $B$ be a real separable Banach space with norm $\parallel \cdot \parallel$ and $\left\{\Omega ,\mathcal{ℱ},P\right\}$ be a probability space. A random element $X$ taking values in $B$ is defined as a Borel measurable function from $\left\{\Omega ,\mathcal{ℱ}\right\}$ into $B$ with Borel sigma-algebra. The expected value of a $B$-valued random variable $X$ is defined to be Bochner integral and is denoted by $EX$.

A Banach space is said to be of Rademacher type $p$, $1\le p\le 2$ if there is a constant $C>0$ such that

for all $n\ge 1$ and each sequence $\left\{X_n,n\ge 1\right\}$ of independent mean zero random elements taking values in $B$ with finite $p$th moments.

We know if $B$ is of Rademacher type $p>1$, then for each $r,1\le r\le p$, $B$ is of Rademacher type $r$. Every separable Hilbert space and finite dimensional Banach space is of Rademacher type 2.

The interested reader can find the complete discussion of this and subsequent notions connected with the geometry of Banach spaces in the book by Ledoux and Talagrand (1991).

A double infinite sequence of random elements $\left\{Y_i,-\infty <i<\infty \right\}$ is said to be stochastically dominated by a random variable $X$ if there exists a constant $C$ such that

for all $x>0$. In the following, $C$ will be used to denote various positive constants.

When $B$ is of Rademacher type $p$, $1<p\le 2$, Shao (1988) showed the following inequality for each sequence $\left\{X_n,n\ge 1\right\}$ of independent,mean zero random elements taking values in $B$ with finite $q$th moments ($q\ge p$)

where $C$ is as in the definition of Rademacher type $p$.

The following lemma (see Lemma 3 of Chow and Lai (1973)) is important for the proof of our second result.

Lemma. Let $\left\{W_n\right\}$ and $\left\{Z_n\right\}$ be two sequences of random variables such that $W_n+Z_n\to 0,a.s.$ Assume that $\left\{{\mathcal{ℱ}}_n\right\}$ is a monotone increasing sequence of $\sigma$-fields. For each $n\ge 1$, $W_1,\cdots ,W_n$ are adapted to $\left\{{\mathcal{ℱ}}_n\right\}$, and $Z_n$ and $\left\{{\mathcal{ℱ}}_n\right\}$ are independent. If $Z_n\to 0$ in probability, then both $W_n$ and $Z_n$ converge to zero almost surely.

3. Main Results

Theorem D. Let $\left\{Y_i,-\infty <i<\infty \right\}$ be a doubly infinite sequence of independent means 0 random elements taking values in a separable real Rademacher type $p\left(1<p\le 2\right)$ Banach space $B$. Assume that $\left\{Y_i,-\infty <i<\infty \right\}$ is stochastically dominated by a real valued random variable $X$. Let $\left\{a_i,-\infty <i<\infty \right\}$ be an absolutely summable sequence of real numbers and set $V_i={\sum }_{k=-\infty }^{\infty }{a}_{i+k}{Y}_k,i\ge 1$ and $S_n={\sum }_{k=1}^n{V}_k$ for $n\ge 1$. If

then

Proof. Let $b={\sum }_{i=-\infty }^{\infty }{a}_i$. Note that

and

Hence for sufficiently large $n$ we have

Let $Y_{nj}=Y_{j}I\left(\parallel Y_j\parallel \le n^{1∕\nu }\right)-E{Y}_{j}I\left(\parallel Y_j\parallel \le n^{1∕\nu }\right)$. Then according to the inequality above, in order to prove the theorem it is enough to prove that

and

For $I_1$ by Chebyshev inequality

For $I_2$, by Chebyshev and Hölder inequalities we have for $q\ge p$

For the case $\left(\beta +2\right)\nu <p$, let $q=p$. By (2.1)

For the case $\left(\beta +2\right)\nu \ge p$, let $q>\frac{\beta -1}{\left(1∕\nu \right)-\left(1∕p\right)}$. By (2.1)

Since $E\mid X{\mid }^{p}I\left(\mid X\mid \le n^{1∕\nu }\right)<\infty$, we have

and by the same argument as $I_2<\infty$ in the case $\left(\beta +2\right)\nu <2$,

The proof of the theorem is completed. $\square$

Corollary. Assume that $\left\{Y_i,-\infty <i<\infty \right\}$ is a doubly infinite sequence of independent means 0 random elements taking values in a separable real Rademacher type $p\left(1<p\le 2\right)$ Banach space $B$ and is stochastically dominated by a real valued random variable $X$. Let $\left\{a_i,-\infty <i<\infty \right\}$ be an absolutely summable sequence of real numbers and set $V_i={\sum }_{k=-\infty }^{\infty }{a}_{i+k}{Y}_k,i\ge 1$ and $S_n={\sum }_{k=1}^n{V}_k$ for $n\ge 1$. If $E\mid X{\mid }^{\nu }<\infty$, where $1<\nu <p$, then

Proof. If $E\mid X{\mid }^{\nu }<\infty$, then by Theorem D with $\beta =-1$

Hence for all $\varepsilon >0$

By Borel-Cantelli Lemma,

which implies that $n^{-1∕\nu }{S}_n\to 0\text{ a.s. }\square$

4. Necessary and Sufficient Condition

The following theorem gives us the necessary and sufficient for the almost sure convergence of partial sums of moving average process.

Theorem E. Let $\left\{Y,Y_i,-\infty <i<\infty \right\}$ be a double infinite sequence of independent identically distributed random elements taking values in a separable real Rademacher type $p\left(1<p\le 2\right)$ Banach space $B$ and $\left\{a_i,-\infty <i<\infty \right\}$ be an absolutely summable sequence of real numbers with ${\sum }_{i=-\infty }^{\infty }{a}_i\ne 0$ and $1<\nu <p$. Let $V_i={\sum }_{k=-\infty }^{\infty }{a}_{i+k}{Y}_k,i\ge 1$ and $S_n={\sum }_{k=1}^n{V}_k$ for $n\ge 1$. Then $n^{-1∕\nu }{S}_n\to 0$ a.s. if and only if

Proof. Note that the sufficiency was proved in the corollary. Hence, we should prove only the necessity part.

Assume that $n^{-1∕\nu }{S}_n\to 0$ a.s. Then

Without loss of generality, we assume that $a_0\ne 0$.

Let $Y^{\prime }$ and $Y_i^{\prime }$ be independent copies of $Y$ and $Y_i,-\infty <i<\infty$, which are also independent of each other. Set $V_i^{\prime }={\sum }_{k=-\infty }^{\infty }{a}_{i+k}{Y}_k^{\prime },i\ge 1$, then $n^{-1∕\nu }{V}_n^{\prime }\to 0$ a.s., and hence

Set

Then $W_n+Z_n\to 0$ a.s., hence $W_n+Z_n\to 0$ in probability. By Lévy inequality, $Z_n\to 0$ in probability and it is easy to show that $\sigma \left(W_1,\cdots ,W_n\right)$ and $Z_n$ are independent. By Lemma we have that $Z_n\to 0$ a.s. Repeating the argument again, we have

Since $a_0\ne 0$, by Borel-Cantelli lemma $E\mid \mid Y-Y^{\prime }\mid {\mid }^{\nu }<\infty$, that is $E\mid \mid Y\mid {\mid }^{\nu }<\infty$.

Because $E\mid \mid Y\mid {\mid }^{\nu }<\infty$, by Theorem D (sufficiency part of the current result), we obtain

By $n^{-1∕\nu }{S}_n\to 0$ a.s. and ${\sum }_{i=-\infty }^{\infty }{a}_i\ne 0$, we have that $EY=0$. $\square$

Remark. It is interesting to find a different proof of the fact that $EY=0$ in the necessity part of Theorem E that is not based on the sufficiency part. We expect that a geometry of the underlying Banach space does not play any role in the necessity part.

5. One additional result.

The following theorem was proved in Baum and Katz (1965).

Theorem F. Let $\left\{X_n,n\ge 1\right\}$ be a sequence of independent identically distributed random variables, $S_n={\sum }_{k=1}^n{X}_k$, and $0<q<2$. Then the conditions $E\mid X_1{\mid }^{\left(\beta +2\right)\nu }<\infty$ and $E{X}_1=0$ for the case $\nu \ge 1$, and $\beta >-1$ are necessary and sufficient for

For $\beta >-1$, Theorem D provides the following extension of Theorem F for the moving average process.

Theorem G. Assume that $\left\{Y_i,-\infty <i<\infty \right\}$ is a doubly infinite sequence of independent means 0 random elements taking values in a separable real Rademacher type $p\left(1<p\le 2\right)$ Banach space $B$ and is stochastically dominated by a real valued random variable $X$. Let $\left\{a_i,-\infty <i<\infty \right\}$ be an absolutely summable sequence of real numbers and set $V_i={\sum }_{k=-\infty }^{\infty }{a}_{i+k}{Y}_k,i\ge 1$ and $S_n={\sum }_{k=1}^n{V}_k$ for $n\ge 1$. If

then

Proof. By Theorem D

Next, we have the following estimations:

Acknowledgment.

The work of Pingyan Chen is supported by the National Natural Science Foundation of China. The work of A.Volodin is supported by a grant from the National Sciences and Engineering Research Council of Canada. The authors are exceptionally grateful to the referee for offering helpful remarks and comments that improved presentation of the paper.

References

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

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

Department of Mathematics, Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China

E-mail address: tchu@math.nthu.edu.tw

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

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

Received March 28, 2006