Open Access

Convergence Theorems for Partial Sums of Arbitrary Stochastic Sequences

Journal of Inequalities and Applications20102010:168081

https://doi.org/10.1155/2010/168081

Received: 27 May 2010

Accepted: 20 October 2010

Published: 24 October 2010

Abstract

By using Doob's martingale convergence theorem, this paper presents a class of strong limit theorems for arbitrary stochastic sequence. Chow's two strong limit theorems for martingale-difference sequence and Loève's and Petrov's strong limit theorems for independent random variables are the particular cases of the main results.

Keywords

Interested ReaderConvergence TheoremPositive Real NumberIndependent Random VariableInteger Number

1. Introduction

Let be a stochastic sequence on the probability space that is, the sequence of -fields in is increasing in (that is ), and are adapted to random variables .

Almost sure behavior of partial sums of random variables has enjoyed both a rich classical period and a resurgence of research activity. Some famous researchers, such as Borel, Kolmogorov, Khintchine, Loève, Chung, and so on, were interested in convergence theorem of partial sums of random variables and obtained lots of classical results for sequences of independent random variables and martingale differences. For a detailed survey of strong limit theorems of sequences for random variables, interested readers can refer to the books [1, 2].

In recent years, some work has been done on the strong limit theorems for arbitrary stochastic sequences. Liu and Yang [3] established two strong limit theorems for arbitrary stochastic sequences, which generalized Chung's [4] strong law of large numbers for sequence of independent random variables as well as Chow's [5] strong law of large numbers for sequence of martingale differences. Then, Yang [6] established two more general strong limit theorems in 2007, which generalized a result by Jardas et al. [7] for sequences of independent random variables and the results by Liu and Yang [3] for arbitrary stochastic sequences in 2003. In 2008, W. Yang and X. Yang [8] proved two strong limit theorems for stochastic sequences, which generalized results by Freedman [9], Isaac, [10] and Petrov [2]. Qiu and Yang [11] established another type strong limit theorem for stochastic sequence in 1999. Then, Wang and Guo [12] extended the main result of Qiu and Yang in 2009. In addition, Wang and Yang [13] established a strong limit theorem for arbitrary stochastic sequences in 2005, which generalized Chow's [5] series convergence theorem for sequence of martingale differences. Then, Qiu [14] extended the result of Wang and Yang in 2008.

The purpose of this paper is to discuss further the strong limit theorems for arbitrary stochastic sequences. By using Doob's [1] convergence theorem for martingale-difference sequence, we establish a class of new strong limit theorems for stochastic sequences. Chow's two strong limit theorems for martingale-difference sequence, Loève's series convergence theorem, and Petrov's strong law of large numbers for sequences of independent random variables are the particular cases of this paper. In addition, the main theorems of this paper extend the main results by Wang and Guo in 2009, Qiu and Yang in 1999, and the result by Wang and Yang in 2005, respectively. The remainder of this paper is organized as follows. In Section 2, we present the main theorems of this paper. In Section 3, the proofs of the main theorems in this paper are presented.

2. Main Theorems

In this section, we will introduce the main results of this paper.

Let be a positive real numbers sequence and and two positive real-valued functions on satisfying when and when .

Theorem 2.1.

Let be a stochastic sequence defined as in Section 1 and a sequence of nondecreasing and nonnegative Borel functions on . For some , suppose that
(2.1)
where and defined as above. Assume that
(2.2)
Set
(2.3)
(i)If there exists some such that
(2.4)
holds when , then
(2.5)
(ii)If there exists some such that (2.4) holds when , then
(2.6)

Corollary 2.2 (Chow).

Let be a martingale-difference sequence and be an increasing sequence of positive numbers. For , let
(2.7)
If , then
(2.8)

Proof.

By using Kroncker's lemma, it is a special case of Theorem 2.1 when the random variables are replaced by and .

Theorem 2.3.

Let be a sequence of arbitrary random variables. Let and . Let and be defined as Theorem 2.1. If
(2.9)

then and converge a.e. under the same conditions (i) and (ii) as in Theorem 2.1, respectively.

Corollary 2.4 (Loève).

Let be a sequence of independent random variables, and . Suppose that
(2.10)

If , then If , then converges a.e.

Corollary 2.5 (Petrov).

Let be a sequence of independent random variables. If and
(2.11)

then when , and when .

Theorem 2.6.

Let be a stochastic sequence defined as in Section 1 and a sequence of nondecreasing and nonnegative Borel functions with on . Let be defined as Theorem 2.1 and
(2.12)
where is a sequence of positive real numbers. Set
(2.13)

Under the same conditions (i) and (ii) as in Theorem 2.1, and converge a.e. on , respectively.

Remark 2.7.

By using Kronecker's lemma, if , we have
(2.14)
(2.15)

respectively.

Theorem 2.8.

Let be a stochastic sequence defined as in Section 1, a sequence of nonzero random variables such that is -measurable, and a sequence of real numbers. Let be two sequences of nonnegative Borel functions on . Suppose that for , does not decrease as , and for ,
(2.16)
holds. Let
(2.17)
Then,
(2.18)
Furthermore, if , one has
(2.19)

Corollary 2.9 (Chow).

Let be a sequence of martingale differences, and let be a sequence of positive real numbers with . For , let
(2.20)
Then,
(2.21)

Corollary 2.10.

Let be an arbitrary stochastic sequence. For , let
(2.22)
Then,
(2.23)

where the log is to the base 2.

Proof.

It is a special case of Theorem 2.8 when , , and (here, we set ).

3. Proofs of Theorems

We first give a lemma.

Lemma 3.1 (see [1]).

Let be a martingale. Then, for some , converges a.e. on the set .

Proof of Theorem 2.1.

Let and a positive integer number. Let ,
(3.1)
where , if the right-hand side of (18) is empty. Then, . Since is measurable , and is nonnegative, we have
(3.2)
Since , we have by (3.2)
(3.3)
By (2.1), (2.2), and (3.3), we obtain
(3.4)
It follows from Borel-Cantelli lemma and (3.4) that holds. Hence, we have
(3.5)
Since , it follows from (3.5) that
(3.6)
Let
(3.7)
It is clear that is a sequence for martingale difference. By using Cr inequality, we have
(3.8)
By using (2.1) and (2.2), we have
(3.9)
Thus, the following inequality holds from (2.3), (3.8), and (3.9)
(3.10)
By using Lemma 3.1, we obtain
(3.11)
Hence, it follows from (3.6), (3.7), and (3.11) that
(3.12)

The following argument breaks down into two cases.

Case 1.

If there exists some such that (2.4) holds when , then
(3.13)
By using (2.3) and (3.13), we obtain
(3.14)

It follows from (3.12) and (3.14) that (2.5) holds.

Case 2.

If there exists some such that the inequality (2.4) holds when , then
(3.15)
By using (2.3) and (3.15), we obtain that
(3.16)

It follows from (3.12) and (3.16) that (2.6) holds. The theorem is proved.

Proof of Theorem 2.3.

Since , we have by (2.9)
(3.17)
It follows from the nonnegative property of that
(3.18)

That is . By Theorem 2.1, the conclusion of Theorem 2.3 holds. The theorem is proved.

Proof of Theorem 2.6.

It is a similar way with Theorem 2.1 except .

Proof of Theorem 2.8.

For , let . Then, is a martingale-difference sequence. It follows from and Jensen's inequality that
(3.19)
Furthermore,
(3.20)
It follows from (2.16) that
(3.21)
holds when . By (3.21), we have
(3.22)
Note that , it follows from (3.19), (3.21), and (3.22) that
(3.23)
And it follows from (2.17) and (3.23) that
(3.24)

It follows from Lemma 3.1 that (2.18) holds. Furthermore, when , it follows from Kronecker's lemma that (2.19) holds. The theorem is proved.

Declarations

Acknowledgment

This work was supported by National Natural Science Foundation of China no. 11071104 and Hebei Natural Science Foundation no. F2010001044.

Authors’ Affiliations

(1)
College of Science, Hebei University of Engineering, Handan, China

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Copyright

© Xiaosheng Wang and Haiying Guo. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.