Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 741901, 7 pages

http://dx.doi.org/10.1155/2013/741901

Research Article

Complete Convergence of the Maximum Partial Sums for Arrays of Rowwise of AANA Random Variables

Aiting Shen, Ranchao Wu, Yan Chen, and Yu Zhou

School of Mathematical Science, Anhui University, Hefei 230601, China

Received 18 January 2013; Revised 1 April 2013; Accepted 4 April 2013

Academic Editor: Garyfalos Papaschinopoulos

Copyright © 2013 Aiting Shen 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.

The limiting behavior of the maximum partial sums is investigated, and some new results are obtained, where is an array of rowwise AANA random variables and is a sequence of positive real numbers. As an application, the Chung-type strong law of large numbers for arrays of rowwise AANA random variables is obtained. The results extend and improve the corresponding ones of Hu and Taylor (1997) for arrays of rowwise independent random variables.

1. Introduction

Let be a sequence of random variables defined on a fixed probability space with value in a real space . We say that the sequence satisfies the general strong law of large numbers if there exist some increasing sequence and some sequence such that

For the definition of general strong law of large numbers, one can refer to Chow and Teicher [1] or Kuczmaszewska [2]. Many authors have extended the strong law of large numbers for sequences of random variables to the case of triangular array of rowwise random variables and arrays of rowwise random variables. In the case of independence, Hu and Taylor [3] proved the following strong law of large numbers.

Theorem A. Let be a triangular array of rowwise independent random variables. Let be a sequence of positive real numbers such that . Let be a positive, even function such that is an increasing function of and is a decreasing function of , respectively, that is,

for some nonnegative integer . If and

where is a positive integer, then

Gan and Chen [4] extended and improved the result of Theorem A for rowwise independent random variable arrays to the case of rowwise negatively associated (NA, in short) random variable arrays. For the details about negatively associated random variables, one can refer to Joag-Dev and Proschan [5], Wu and Jiang [6, 7], Wu [8], Wang et al. [9], and so forth. In this paper, we will further study the strong law of large numbers for arrays of rowwise asymptotically almost negatively associated random variables based on some different conditions.

The concept of asymptotically almost negatively associated random variables was introduced by Chandra and Ghosal [10] as follows.

Definition 1. A sequence of random variables is called asymptotically almost negatively associated (AANA, in short) if there exists a nonnegative sequence as such that

for all , and for all coordinatewise nondecreasing continuous functions and whenever the variances exist.

An array of random variables is called rowwise AANA random variables if for every , is a sequence of AANA random variables.

It is easily seen that the family of AANA sequence contains NA (in particular, independent) sequences (with , ) and some more sequences of random variables which are not much deviated from being negatively associated. An example of an AANA sequence which is not NA was constructed by Chandra and Ghosal [10]. Hence, extending the limit properties of independent or NA random variables to the case of AANA random variables is highly desirable in the theory and application.

Since the concept of AANA sequence was introduced by Chandra and Ghosal [10], many applications have been found. See, for example, Chandra and Ghosal [10] who derived the Kolmogorov-type inequality and the strong law of large numbers of Marcinkiewicz-Zygmund; Chandra and Ghosal [11] obtained the almost sure convergence of weighted averages; Wang et al. [12] established the law of the iterated logarithm for product sums; Ko et al. [13] studied the Hájek-Rényi-type inequality; Yuan and An [14] established some Rosenthal-type inequalities for maximum partial sums of AANA sequence; Wang et al. [15] obtained some strong growth rate and the integrability of supremum for the partial sums of AANA random variables; Wang et al. [16] established some maximal inequalities and strong law of large numbers for AANA sequences; Wang et al. [17, 18] studied complete convergence for arrays of rowwise AANA random variables and weighted sums of arrays of rowwise AANA random variables, respectively; Hu et al. [19] studied the strong convergence properties for AANA sequence; Yang et al. [20] investigated the complete convergence, complete moment convergence, and the existence of the moment of supremum of normed partial sums for the moving average process for AANA sequence; and so forth.

The main purpose of this paper is to study the limiting behavior of the maximum partial sums for arrays of rowwise AANA random variables. As an application, the Chung-type strong law of large numbers for arrays of rowwise AANA random variables is obtained. We will give some sufficient conditions for the complete convergence for an array of rowwise AANA random variables without assumptions of identical distribution and stochastic domination. The results presented in this paper are obtained by using the truncated method and the maximal Rosenthal-type inequality of AANA random variables.

Throughout this paper, let be an array of rowwise of AANA random variables with the mixing coefficients in each row. The symbol denotes a positive constant which may be different in various places.

2. Preliminaries

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

Lemma 2 (see [18, Lemma 2.2]). Let be a sequence of AANA random variables with mixing coefficients , and let be all nondecreasing (or all nonincreasing) continuous functions, and then is still a sequence of AANA random variables with mixing coefficients .

Lemma 3 (see [14, Theorem 2.1]). Let and let be a sequence of zero mean random variables with mixing coefficients .

If , then there exists a positive constant depending only on such that for all and ,

If   for some , where the integer number , then there exists a positive constant depending only on such that for all ,

3. Main Results and Proofs

In this section, we will investigate the limiting behavior of the maximum partial sums for arrays of rowwise of AANA random variables. The first three theorems consider different conditions from Hu and Taylor’s [3].

Theorem 4. Let be an array of rowwise AANA random variables with and let be a sequence of positive real numbers. Let be a sequence of nonnegative, even functions such that is an increasing function of for every . Assume that there exists a constant such that for . If

then for any ,

Proof. For fixed , define

By Lemma 2, we can see that for fixed , is still a sequence of AANA random variables. It is easy to check that for any ,

which implies that

Firstly, we will show that

In fact, by the conditions for and (8), we have

which implies (13).

It follows from (12) and (13) that for large enough,

Hence, to prove (9), we only need to show that

When , we have , which yields that

Hence,

which implies (16).

By Markov’s inequality, Lemma 3 (for ), for , and (8), we can get that

which implies (17). This completes the proof of the theorem.

Corollary 5. Let be an array of rowwise AANA random variables with and let be a sequence of positive real numbers. If there exists a constant such that

then (9) holds for any .

Proof. In Theorem 4, we take

It is easy to check that is a sequence of nonnegative, even functions such that is an increasing function of for every . And

Therefore, by Theorem 4, we can easily get (9).

Corollary 6. Under the conditions of Theorem 4 or Corollary 5,

Theorem 7. Let be an array of rowwise AANA random variables with and let be a sequence of positive real numbers. , , . Let be a sequence of nonnegative, even functions. Assume that there exist and such that for and there exists a such that for . If (8) satisfies, then (9) holds for any .

Proof. We use the same notations as those in Theorem 4. The proof is similar to that of Theorem 4.

Firstly, we will show that (13) holds true. Actually, by the conditions , for , and (8), we have

which implies (13). Hence, to prove (9), we only need to show that (16) and (17) hold true.

The conditions for and (8) yield that

which implies (16).

By Markov’s inequality, Lemma 3 (for ), for , , and (8), we can get that

which implies (17). This completes the proof of the theorem.

Corollary 8. Let be an array of rowwise AANA random variables with and let be a sequence of positive real numbers. , , . If there exists a constant such that

then (9) holds for any .

Proof. In Theorem 7, we take

It is easy to check that is a sequence of nonnegative, even functions satisfying

Therefore, by Theorem 7, we can easily get (9).

Furthermore, by Corollaries 5 and 8, we can get the following important Chung-type strong law of large numbers for arrays of rowwise AANA random variables.

Corollary 9. Let be an array of rowwise AANA random variables with and let be a sequence of positive real numbers. If there exists some such that

and , , if , then (9) holds for any and .

For , we have the following result.

Theorem 10. Let be an array of rowwise AANA random variables with and let be a sequence of positive real numbers. Let be a sequence of nonnegative, even functions. Assume that there exists some such that for . If

then (9) holds for any and .

Proof. We use the same notations as those in Theorem 4. The proof is similar to that of Theorem 4. It is easily seen that (32) implies that

Firstly, we will show that (13) holds true. In fact, by Hölder’s inequality, for , (32), and (33), we have

which implies (13). To prove (9), we only need to show that (16) and (17) hold true.

By the condition for again and (33), we have

which implies (16).

By Markov’s inequality, Lemma 3 (for ), for , and (34), we can get that

which implies (17). This completes the proof of the theorem.

The next two theorems extend and improve Theorem A for arrays of rowwise of independent random variables to the case of AANA random variables.

Theorem 11. Let be an array of rowwise AANA random variables with and let be a sequence of positive real numbers. Let be a sequence of nonnegative and even functions such that for and every , and

where . Assume further that

Then (9) holds for any and    .

Proof. We use the same notations as those in Theorem 4. The proof is similar to that of Theorem 4.

Firstly, we will show that (13) holds true. Actually, by the conditions and (38)–(40), we have

which implies (13). Hence, to prove (9), it suffices to show that (16) and (17) hold true.

The conditions (38) and (40) yield that

which implies (16).

By Markov’s inequality, Lemma 3, (38), and (40), we can get that

which implies (17). This completes the proof of the theorem.

Theorem 12. Let be an array of rowwise AANA random variables and let be a sequence of positive real numbers. Let , , , and , where the integer number . Let be a sequence of nonnegative and even functions such that for and every . Assume that conditions (38)–(40) are satisfied. If and

then (9) holds for any and .

Proof. We use the same notations as those in Theorem 4. The proof is similar to that of Theorems 4 and 11.

Following the methods of the proof in Theorems 4 and 11, (13) and (16) hold. So we need only to show (17). Denote We have by Markov’s inequality, Lemma 3, and ’s inequality that

Since and , as , it follows that and as . Hence,

It follows by , , , and (44) that

Therefore, (17) follows from (45)–(47) immediately. This completes the proof of the theorem.

Acknowledgments

The authors are most grateful to the Editor and anonymous referee 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 (11201001, 11171001, and 11226207), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20093401120001), the Natural Science Foundation of Anhui Province (1308085QA03, 11040606M12, and 1208085QA03), the 211 project of Anhui University, the Youth Science Research Fund of Anhui University, Applied Teaching Model Curriculum of Anhui University (XJYYXKC04), and the Students Science Research Training Program of Anhui University (KYXL2012007).

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