- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Discrete Dynamics in Nature and Society

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

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

## Strong and Weak Convergence for Asymptotically Almost Negatively Associated Random Variables

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).

#### References

- J. Cuzick, “A strong law for weighted sums of i.i.d. random variables,”
*Journal of Theoretical Probability*, vol. 8, no. 3, pp. 625–641, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Bai, P. E. Cheng, and C.-H. Zhang, “An extension of the Hardy-Littlewood strong law,”
*Statistica Sinica*, vol. 7, no. 4, pp. 923–928, 1997. View at Zentralblatt MATH · View at MathSciNet - Z. Bai and P. Cheng, “Marcinkiewicz strong laws for linear statistics,”
*Statistics & Probability Letters*, vol. 46, no. 2, pp. 105–112, 2000. - G.-H. Cai, “Marcinkiewicz strong laws for linear statistics of ${\rho}^{\ast}$-mixing sequences of random variables,”
*Anais da Academia Brasileira de Ciências*, vol. 78, no. 4, pp. 615–621, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - Q. Wu, “A strong limit theorem for weighted sums of sequences of negatively dependent random variables,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 383805, 8 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. H. Sung, “On the strong convergence for weighted sums of random variables,”
*Statistical Papers*, vol. 52, no. 2, pp. 447–454, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X.-C. Zhou, C.-C. Tan, and J.-G. Lin, “On the strong laws for weighted sums of ${\rho}^{\ast}$-mixing random variables,”
*Journal of Inequalities and Applications*, vol. 2011, Article ID 157816, 8 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - X. Wang, X. Li, W. Yang, and S. Hu, “On complete convergence for arrays of rowwise weakly dependent random variables,”
*Applied Mathematics Letters*, vol. 25, no. 11, pp. 1916–1920, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. K. Chandra and S. Ghosal, “Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables,”
*Acta Mathematica Hungarica*, vol. 71, no. 4, pp. 327–336, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. K. Chandra and S. Ghosal, “The strong law of large numbers for weighted averages under dependence assumptions,”
*Journal of Theoretical Probability*, vol. 9, no. 3, pp. 797–809, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Wang, J. Yan, F. Cheng, and C. Su, “The strong law of large numbers and the law of the iterated logarithm for product sums of NA and AANA random variables,”
*Southeast Asian Bulletin of Mathematics*, vol. 27, no. 2, pp. 369–384, 2003. View at Zentralblatt MATH · View at MathSciNet - M.-H. Ko, T.-S. Kim, and Z. Lin, “The Hájeck-Rènyi inequality for the AANA random variables and its applications,”
*Taiwanese Journal of Mathematics*, vol. 9, no. 1, pp. 111–122, 2005. View at MathSciNet - D. Yuan and J. An, “Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications,”
*Science in China. Series A*, vol. 52, no. 9, pp. 1887–1904, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Wang, S. Hu, and W. Yang, “Convergence properties for asymptotically almost negatively associated sequence,”
*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 218380, 15 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Wang, S. Hu, and W. Yang, “Complete convergence for arrays of rowwise asymptotically almost negatively associated random variables,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 717126, 11 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Wang, S. Hu, W. Yang, and X. Wang, “On complete convergence of weighted sums for arrays of rowwise asymptotically almost negatively associated random variables,”
*Abstract and Applied Analysis*, Article ID 315138, 15 pages, 2012. View at MathSciNet - X. P. Hu, G. H. Fang, and D. J. Zhu, “Strong convergence properties for asymptotically almost negatively associated sequence,”
*Discrete Dynamics in Nature and Society*, vol. 2012, Article ID 562838, 8 pages, 2012. View at Publisher · View at Google Scholar - W. Yang, X. Wang, N. Ling, and S. Hu, “On complete convergence of moving average process for AANA sequence,”
*Discrete Dynamics in Nature and Society*, vol. 2012, Article ID 863931, 24 pages, 2012. View at Zentralblatt MATH · View at MathSciNet - W. Feller, “A limit theorem for random variables with infinite moments,”
*American Journal of Mathematics*, vol. 68, no. 2, pp. 257–262, 1946.