Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 235012, 7 pages

http://dx.doi.org/10.1155/2013/235012

Research Article

Strong and Weak Convergence for Asymptotically Almost Negatively Associated Random Variables

Aiting Shen and Ranchao Wu

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

Received 10 December 2012; Accepted 16 January 2013

Academic Editor: Binggen Zhang

Copyright © 2013 Aiting Shen and Ranchao Wu. 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 strong law of large numbers for sequences of asymptotically almost negatively associated (AANA, in short) random variables is obtained, which generalizes and improves the corresponding one of Bai and Cheng (2000) for independent and identically distributed random variables to the case of AANA random variables. In addition, the Feller-type weak law of large number for sequences of AANA random variables is obtained, which generalizes the corresponding one of Feller (1946) for independent and identically distributed random variables.

1. Introduction

Many useful linear statistics based on a random sample are weighted sums of independent and identically distributed random variables. Examples include least-squares estimators, nonparametric regression function estimators, and jackknife estimates,. In this respect, studies of strong laws for these weighted sums have demonstrated significant progress in probability theory with applications in mathematical statistics.

Let be a sequence of random variables and let be an array of constants. A common expression for these linear statistics is . Some recent results on the strong law for linear statistics can be found in Cuzick [1], Bai et al. [2], Bai and Cheng [3], Cai [4], Wu [5], Sung [6], Zhou et al. [7], and Wang et al. [8]. Our emphasis in this paper is focused on the result of Bai and Cheng [3]. They gave the following theorem.

Theorem A. Suppose that , , and . Let be a sequence of independent and identically distributed random variables satisfying , and let be an array of real constants such that

If , then

We point out that the independence assumption is not plausible in many statistical applications. So it is of interest to extend the concept of independence to the case of dependence. One of these dependence structures is asymptotically almost negatively associated, which was introduced by Chandra and Ghosal [9] 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.

It is easily seen that the family of AANA sequence contains negatively associated (NA, in short) 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 [9]. 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 [9], many applications have been found. See, for example, Chandra and Ghosal [9] derived the Kolmogorov type inequality and the strong law of large numbers of Marcinkiewicz-Zygmund; Chandra and Ghosal [10] obtained the almost sure convergence of weighted averages; Wang et al. [11] established the law of the iterated logarithm for product sums; Ko et al. [12] studied the Hájek-Rényi type inequality; Yuan and An [13] established some Rosenthal type inequalities for maximum partial sums of AANA sequence; Wang et al. [14] obtained some strong growth rate and the integrability of supremum for the partial sums of AANA random variables; Wang et al. [15, 16] studied complete convergence for arrays of rowwise AANA random variables and weighted sums of arrays of rowwise AANA random variables, respectively; Hu et al. [17] studied the strong convergence properties for AANA sequence; Yang et al. [18] investigated the complete convergence, complete moment convergence, and the existence of the moment of supermum 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 strong convergence for AANA random variables, which generalizes and improves the result of Theorem A. In addition, we will give the Feller-type weak law of large number for sequences of AANA random variables, which generalizes the corresponding one of Feller [19] for independent and identically distributed random variables.

Throughout this paper, let be a sequence of AANA random variables with the mixing coefficients . . For , let be the dual number of . The symbol denotes a positive constant which may be different in various places. Let be the indicator function of the set . stands for .

The definition of stochastic domination will be used in the paper as follows.

Definition 2. 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 3. Suppose that , , and . Let be a sequence of AANA random variables, which is stochastically dominated by a random variable and , if . Suppose that there exists a positive integer such that for some and . Let be an array of real constants satisfying

If , then

Remark 4. Theorem 3 generalizes and improves Theorem A of Bai and Cheng [3] for independent and identically distributed random variables to the case of AANA random variables, since Theorem 3 removes the identically distributed condition and expands the ranges , , and , respectively.

At last, we will present the Feller-type weak law of large number for sequences of AANA random variables, which generalizes the corresponding one of Feller [19] for independent and identically distributed random variables.

Theorem 5. Let and be a sequence of identically distributed AANA random variables with the mixing coefficients satisfying . If

then

2. Preparations

To prove the main results of the paper, we need the following lemmas. The first two lemmas were provided by Yuan and An [13].

Lemma 6 (cf. see [13, Lemma 2.1]). Let be a sequence of AANA random variables with mixing coefficients , be all nondecreasing (or all nonincreasing) continuous functions, then is still a sequence of AANA random variables with mixing coefficients .

Lemma 7 (cf. see [13, Theorem 2.1]). Let and 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 integer number , then there exists a positive constant depending only on such that for all ,

The last one is a fundamental property for stochastic domination. The proof is standard, so the details are omitted.

Lemma 8. Let be a sequence of random variables, which is stochastically dominated by a random variable . Then for any and ,

where and are positive constants.

3. Proofs of the Main Results

Proof of Theorem 3. Without loss of generality, we assume that (otherwise, we use and instead of , and note that ). Denote for and that

Hence, , which implies that

To prove (6), it suffices to show that ., and . as .

Firstly, we will show that .

For any , it follows from (5) and Hölder's inequality that

for any , it follows from (5) again that

Combining (14) and (15), we have

The condition yields that

which implies that by Borel-Cantelli lemma. Thus, we have by (5) that

Secondly, we will prove that

If , then we have by Lemma 8 and (16) that

If , then we have by , Lemma 8 and (16) that

Hence, (19) follows from (20) and (21) immediately.

To prove (6), it suffices to show that

By Borel-Cantelli Lemma, we only need to show that for any ,

For fixed , it is easily seen that are still AANA random variables by Lemma 6. Taking , we have by Markov's inequality and Lemma 7 that

For , we have by inequality, Jensen's inequality, (15), and Lemma 8 that

Next, we will prove that . By inequality, Jensen's inequality and Lemma 8 again, we can see that

It follows by Markov's inequality and the fact that

If we denote , then we can get by (26) and (27) that

It is easily seen that

Hence, we have by (28) and (29) that

which together with yields (23). This completes the proof of the theorem.

Proof of Theorem 5. Denote for and that

and . By the assumption (7), we have for any that

which implies that

Hence, in order to prove (8), we only need to show that

By (7) again and Toeplitz's lemma, we can get that

Note that

Combing (35) and (36), we have

By Lemma 7 (taking ), (7), and (37), we can get that

This completes the proof of the theorem.

Acknowledgments

The authors are most grateful to the Editor Binggen Zhang and anonymous referee for the 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 11126176), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20093401120001), the Natural Science Foundation of Anhui Province (11040606M12, 1208085QA03), the Natural Science Foundation of Anhui Education Bureau (KJ2010A035), the 211 project of Anhui University, the Academic Innovation Team of Anhui University (KJTD001B), and the Students Science Research Training Program of Anhui University (KYXL2012007).

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