Abstract and Applied Analysis

Volume 2012 (2012), Article ID 572493, 13 pages

doi:10.1155/2012/572493

Research Article

Convergence Rates in the Strong Law of Large Numbers for Martingale Difference Sequences

Xuejun Wang, Shuhe Hu, Wenzhi Yang, and Xinghui Wang

School of Mathematical Science, Anhui University, Hefei 230039, China

Received 31 May 2012; Accepted 14 June 2012

Academic Editor: Sung Guen Kim

Copyright © 2012 Xuejun Wang et al. 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.

Abstract.

We study the complete convergence and complete moment convergence for martingale difference sequence. Especially, we get the Baum-Katz-type Theorem and Hsu-Robbins-type Theorem for martingale difference sequence. As a result, the Marcinkiewicz-Zygmund strong law of large numbers for martingale difference sequence is obtained. Our results generalize the corresponding ones of Stoica (2007, 2011).

1. Introduction

The concept of complete convergence was introduced by Hsu and Robbins [1] as follows. A sequence of random variables is said to converge completely to a constant if for all . In view of the Borel-Cantelli lemma, this implies that almost surely (a.s.). The converse is true if the are independent. Hsu and Robbins [1] proved that the sequence of arithmetic means of independent and identically distributed (i.i.d.) random variables converges completely to the expected value if the variance of the summands is finite. Erdös [2] proved the converse. The result of Hsu-Robbins-Erdös is a fundamental theorem in probability theory and has been generalized and extended in several directions by many authors. One of the most important generalizations is Baum and Katz [3] for the strong law of large numbers as follows.

Theorem A (see Baum and Katz [3]). Let and let . Let be a sequence of independent and identically distributed random variables. Assume further that if . Then the following statements are equivalent:

(i),(ii) for all .

Motivated by Baum and Katz [3] for independent and identically distributed random variables, many authors studied the Baum-Katz-type Theorem for dependent random variables; see, for example, -mixing random variables, -mixing random variables, negatively associated random variables, martingale difference sequence, and so forth.

Our emphasis in the paper is focused on the Baum-Katz-type Theorem for martingale difference sequence. Recently, Stoica [4, 5] considered the following series that describes the rate of convergence in the strong law of large numbers:

They obtained the follow results.

Theorem B (see Stoica [4]). Let be an -bounded martingale difference sequence, and let . Then series (1.1) converges for all .

Theorem C (see Stoica [5]). (i) Let , and let . Then the series (1.1) converges for any martingale difference sequence bounded in .

(ii) Let and . Then the series (1.1) converges for any martingale difference sequence satisfying .

The main purpose of the paper is to further study the Baum-Katz-type Theorem for martingale difference sequence. We have the following generalizations.

(i)Our results include Baum-Katz-type Theorem and Hsu-Robbins-type Theorem (see Hsu and Robbins [1]) as special cases.(ii)Our results generalize Theorems B and C for the partial sum to the case of maximal partial sum.(iii)Our results not only generalize Theorem B for and Theorem C (i) for , to the case of , and but also generalize Theorem C (ii) for to the case of .

Throughout the paper, let be a sequence of random variables defined on a fixed probability space . Denote , , and . stands for . , denote positive constants which may be different in various places. denotes the integer part of . Let be the indicator function of the set .

Let be an increasing sequence of fields with for each . If is measurable for each , then fields are said to be adapted to the sequence , and is said to be an adapted stochastic sequence.

Definition 1.1. If is an adapted stochastic sequence with

and for each , then the sequence is called a martingale difference sequence.

The following two definitions will be used frequently in the paper.

Definition 1.2. A real-valued function , positive and measurable on , is said to be slowly varying if

for each .

Definition 1.3. A sequence of random variables is said to be stochastically dominated by a random variable if there exists a positive constant , such that

for all and .

Our main results are as follows.

Theorem 1.4. Let , and let . Let be a martingale difference sequence, which is stochastically dominated by a random variable . Let be a slowly varying function as . Supposing that if and

then for any ,

Theorem 1.5. Let , and let . Let be a martingale difference sequence, which is stochastically dominated by a random variable . Let be a slowly varying function as . Supposing that if and (1.5) holds, then for any ,

For and , we have the following theorem.

Theorem 1.6. Let , and let be a martingale difference sequence, which is stochastically dominated by a random variable . Supposing that

then for any ,

The following theorem presents the complete moment convergence for martingale difference sequence.

Theorem 1.7. Letting the conditions of Theorem 1.4 hold, then for any ,

Remark 1.8. If we take in Theorem 1.4, then we can not only get the Baum-Katz-type Theorem for martingale difference sequence but also consider the case of . Furthermore, if we take , , and in Theorem 1.4, then we can get the Hsu-Robbins-type Theorem (see Hsu and Robbins [1]) for martingale difference sequence.

Remark 1.9. As stated above, our Theorems 1.4 and 1.5 not only generalize the corresponding results of Theorems B and C for the partial sum to the maximal partial sum but also expand the scope of and .

Remark 1.10. If we take in Theorem 1.4, then we can get the Marcinkiewicz-Zygmund strong law of large numbers for martingale difference sequence as follows:

2. Preparations

To prove the main results of the paper, we need the following lemmas.

Lemma 2.1 (see [6, Theorem 2.11]). If is a martingale difference and , then there exists a constant depending only on such that

Lemma 2.2. Let be a sequence of random variables, which is stochastically dominated by a random variable . Then for any and , the following two statements hold:

where and are positive constants.

Lemma 2.3 (cf. [7]). If is a slowly varying function as , then

(i) for each ; for each ,(ii),(iii), for each ,(iv) for every , , positive integer and some , ,(v) for every , , positive integer and some , .

3. Proofs of the Main Results

Proof of Theorem 1.4. For fixed , denote

Since , we can see that

For , we have by Markov’s inequality, Lemma 2.2, and (1.5) that

For , we have by Markov’s inequality and (3.3) that

To prove (1.6), it suffices to show that

For fixed , it is easily seen that is still a martingale difference. By Markov’s inequality and Lemma 2.1, we have that for any ,

We consider the following three cases.

Case 1 (and ). Take large enough such that , which implies that .

For , we have by ’s inequality, Lemma 2.2, (3.3), Lemma 2.3, and (1.5) that

Note that , if . We have by Lemma 2.3 that

Case 2 (and ). Take . Similar to the proof of (3.6) and (3.7), we can get that

Case 3 (). Note that . Take , and similar to the proof of (3.9), we still have .

From the statements mentioned previously, we have proved (3.5). This completes the proof of the theorem.

Proof of Theorem 1.5. We have by Lemma 2.3 that

The desired result (1.7) follows from the inequality above and (1.6) immediately.

Proof of Theorem 1.6. We use the same notation as that in Theorem 1.4. According to the proof of Theorem 1.4, we can see that for and under the conditions of Theorem 1.6. So it suffices to show that and for and .

Similar to the proof of (3.3), we have

Similar to the proof of (3.4) and (3.11), we can get that

This completes the proof of the theorem.

Proof of Theorem 1.7. For any , we have by Theorem 1.4 that

Hence, it suffices to show that

For , denote

Since , it follows that

Similar to the proof of (3.3), we have by Markov’s inequality and Lemma 2.2 that

According to the proof of (3.17), we have by Markov’s inequality and Lemma 2.2 that

For any , it is easily seen that is still a martingale difference. By Markov’s inequality and Lemma 2.1, we have that for any ,

We still consider the following three cases.

Case 1 (and ). Take large enough such that , which implies that . We have by Lemma 2.2 and (3.17) that

Hence, similar to the proof of (3.7), we can see that

Note that , if . We have by Lemma 2.3 that

Case 2 (and ). Take . Similar to the proof of (3.19) and (3.21), we can get that

Case 3 (). Note that . Take , and similar to the proof of (3.23), we still have .

From the statements mentioned previously, we have proved (3.14). This completes the proof of the theorem.

Acknowledgments

The authors are most grateful to the Editor Sung Guen Kim and anonymous referees for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (11171001, 11126176), Natural Science Foundation of Anhui Province (1208085QA03), Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005), Doctoral Research Start-up Funds Projects of Anhui University, the Academic Innovation Team of Anhui University (KJTD001B), and The Talents Youth Fund of Anhui Province Universities (2010SQRL016ZD, 2011SQRL012ZD).

References

P. L. Hsu and H. Robbins, “Complete convergence and the law of large numbers,” Proceedings of the National Academy of Sciences of the United States of America, vol. 33, no. 2, pp. 25–31, 1947.

P. Erdös, “On a theorem of Hsu and Robbins,” Annals of Mathematical Statistics, vol. 20, no. 2, pp. 286–291, 1949.

L. E. Baum and M. Katz, “Convergence rates in the law of large numbers,” Transactions of the American Mathematical Society, vol. 120, no. 1, pp. 108–123, 1965.

G. Stoica, “Baum-Katz-Nagaev type results for martingales,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 1489–1492, 2007.

G. Stoica, “A note on the rate of convergence in the strong law of large numbers for martingales,” Journal of Mathematical Analysis and Applications, vol. 381, no. 2, pp. 910–913, 2011.

P. Hall, Martingale Limit Theory and Its Application, Academic Presss, New York, NY, USA, 1980.

Z. D. Bai and C. Su, “The complete convergence for partial sums of i.i.d. random variables,” Science in China Series A, vol. 28, no. 12, pp. 1261–1277, 1985.