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Abstract and Applied Analysis
Volume 2013, Article ID 295041, 12 pages
http://dx.doi.org/10.1155/2013/295041
Research Article

Strong Convergence Properties and Strong Stability for Weighted Sums of AANA Random Variables

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

Received 10 August 2013; Accepted 22 October 2013

Academic Editor: Jaume Giné

Copyright © 2013 Zhiyong Chen 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 Khintchine-Kolmogorov-type convergence theorem and three-series theorem for AANA random variables are established. By using these convergence theorems, we obtain convergence results for AANA sequences, which extend the corresponding ones for independent sequences and NA sequences. In addition, we study the strong stability for weighted sums of AANA random variables and obtain some new results, which extend some earlier ones for NA random variables.

1. Introduction

Firstly, let us recall some definitions.

Definition 1 (cf. Wu [1]). A sequenceof random variables is said to be stochastically dominated by a random variableif there exists a constant such that for alland.

Definition 2 (cf. Chow and Teicher [2]). A sequenceof random variables is said to be strongly stable if there exist two constant sequencesandwithsuch that

Definition 3 (cf. Wu [1]). A functionis said to be quasimonotonically increasing function if there existand constantwithsuch that. A functionis said to be quasimonotonically decreasing function if there existand constantwithsuch that.

Definition 4 (cf. Wu [1]). A real-valued function, positive and measurable on, is said to be slowly varying if for each.

Definition 5 (cf. Joag-Dev and Proschan [3]). A finite collection of random variablesis said to be negatively associated (NA, in short) if for every pair of disjoint subsets, of, wheneverandare coordinatewise nondecreasing such that this covariance exists. An infinite sequenceis NA if every finite subcollection is NA.

Definition 6 (cf. Chandra and Ghosal [4]). A sequenceof random variables is called asymptotically almost negatively associated (AANA) if there exists a nonnegative sequenceassuch that for alland for all coordinatewise nondecreasing continuous functionsandwhenever the variances exist.

Obviously, the family of AANA sequences contains NA (in particular, independent) sequences (taking, ) and some more sequences of random variables which do not much deviates from being NA. An example of an AANA sequence which is not NA was introduced by Chandra and Ghosal [5].

Since the concept of AANA random variables was introduced by Chandra and Ghosal [4], many applications have been found. For example, Chandra and Ghosal [4] derived the Kolmogorov-type inequality and the strong law of large numbers of Marcinkiewicz-Zygmund, Chandra and Ghosal [5] obtained the almost sure convergence of weighted averages, Ko et al. [6] studied the Hájek-Rènyi type inequality, and Wang et al. [7] established the law of the iterated logarithm for product sums. Yuan and An [8] established Rosenthal-type inequalities for maximum partial sums of AANA sequences. Wang et al. [9] studied some convergence properties for AANA sequence. Wang et al. [10] generalized and improved the results of Ko et al. [6] and studied the large deviation and Marcinkiewicz-type strong law of large numbers for AANA sequences. Yang et al. [11] investigated the complete convergence of moving average process for AANA sequence. Hu et al. [12] and Shen and Wu [13, 14] studied strong convergence property for weighted sums of AANA sequence. Wang et al. [15, 16] and Shen et al. [17] obtained some results on complete convergence for AANA sequence, and so forth.

In this paper, we mainly study convergence results for AANA random variables, and the strong stability for weighted sums of AANA random variables, which extend the corresponding ones for independent sequences and NA sequences without necessarily adding extra conditions. The techniques used in the paper are the truncated method, the Khintchine-Kolmogorov-type convergence theorem and three-series theorem for AANA random variables.

Throughout this paper, letbe the indicator function of the set, andfor some.denotes that there exists a positive constantsuch that. The symbolrepresents a positive constant which may be different in various places. The main results of this paper depend on the following lemmas.

Lemma 7 (cf. Yuan and An [8]). Letbe a sequence of AANA random variables with mixing coefficients, and letbe all nondecreasing (or nonincreasing) continuous functions; thenis still a sequence of AANA random variables with mixing coefficients.

Lemma 8 (cf. Wang et al. [9]). Letandbe a sequence of AANA random variables with mixing coefficients. Assume thatfor alland; then there exists a positive constantdepending only onsuch that for all, where.

By Lemmas 7 and 8, we can get the following Khintchine-Kolmogorov-type convergence theorem and three series theorem for AANA sequences, which can be applied to prove the main results of the paper. The proofs are standard, so we omit them.

Corollary 9 (Khintchine-Kolmogorov-type convergence theorem). Letbe a sequence of AANA random variables with mixing coefficientsand. If thenconverges almost surely.

Corollary 10 (three-series theorem for AANA random variables). Letbe a sequence of AANA random variables with mixing coefficientsand. Assume that for some, Then,converges almost surely.

Remark 11. Since NA implies AANA, Corollaries 9 and 10 extend corresponding results for NA random variables (see Matula [18]) to AANA random variables without adding any extra condition.

Lemma 12 (cf. Wu [19] or Shen [20]). Letbe a sequence of random variables which is stochastically dominated by a random variable. For anyand, the following two statements hold: whereandare positive constants.

Lemma 13 (cf. Wu [1]). Letbe a slowly varying function; then for any,is a quasimonotonically increasing function andis a quasimonotonically decreasing function.

2. Strong Convergence Properties of Weighted Sums for AANA Sequence

Theorem 14. Letbe a sequence of AANA random variables with. Assume thatis a sequence of even functions defined on. For each,is a positive and nondecreasing function inand satisfies one of the following conditions:(i)for some,is a nondecreasing function in;(ii)for some,andare nonincreasing functions in; furthermore, assume thatfor each.
For any positive number sequencewith such that then converges a.s., and

Proof. For each, denote By Lemma 7, we can see that, for fixed,is still a sequence of AANA random variables. So by Corollary 10 in order to prove (11), we need only to prove the convergence of three series of (8), where.
Firstly, we prove that under condition (i) or (ii).
For each, ifsatisfies condition (i), noting thatis a sequence of positive and nondecreasing even function in. Combining Markov’s inequality with (10), it follows that Ifsatisfies condition (ii), it is easy to prove that (13) also holds.
Secondly, we will show.
Ifsatisfies (i), when, we have, which implies that Note thatis a sequence of positive and nondecreasing functions in, sowhen. Consequently, On the other hand, ifsatisfies condition (ii), then we can also get that Therefore, whether even functionsatisfies condition (i) or (ii), we can obtain Therefore, it follows from (10) that
Finally, we prove that.
Ifsatisfies condition (i), when, we have, for. It follows that
Ifsatisfies condition (ii), then by the fact thatandis a nonincreasing function in, we get Therefore, whethersatisfies condition (i) or (ii), it also follows from (10) that The proof of Theorem 14 is completed by (13), (18), and (21).

Corollary 15. Letbe a sequence of AANA random variables with, and letbe a sequence of positive numbers with. There exists somesuch that If, we further assume that. Then, (11) holds.

Proof. We take,. If, one can find thatfor. So condition (i) of Theorem 14 is satisfied. If, we have thatandfor. Thus, condition (ii) of Theorem 14 is satisfied. Consequently, the desired result (11) follows from Theorem 14 immediately.

Remark 16. If takingin (i) andin (ii), Theorem 14 and Corollary 15 extend the corresponding ones for NA random variables (see Gan [21]) to AANA random variables.

Theorem 17. Letand be a sequence of AANA random variables withand identical distribution whereis a slowly varying function. Letandbe sequences of positive constants satisfying. Denotefor each. Assume that then

Proof. Since (23) and (24) imply thatfor all sufficiently large. Without loss of generality, we assumefor all.
By Borel-Cantelli Lemma, it is easily seen that (24) implies that Denote thus,is still AANA from Lemma 7. It is easy to check that In order to show that a.s., we only need to show that the first three terms above areora.s.
Byinequality, Theorem  1b in [22, page 281] (or see Adler [23]) and (24), we can get It follows from Corollary 9 and Kronecker’s lemma that By (24) again, which implies that By Kronecker’s lemma, it follows that By Theorem  1b in [22, page 281] (or see Adler [23]) and (24) again, we have which implies that By Kronecker’s Lemma, it follows that Hence, the desired result (25) follows from (26)–(36) immediately.

Remark 18. Theorem 17 generalizes and extends the corresponding one for NA random variables (see Wang et al. [24]) to AANA random variables.

Theorem 19. Letanda sequence of mean zero AANA random variables with, which is stochastically dominated by a random variable. Letbe a sequence of positive constants satisfying. Denotefor each. Assume that then

Proof. Letand denote It follows by (37) that By the equality above and Borel-Cantelli lemma, we can get. Therefore, in order to prove (38), we only need to prove that
Byinequality, Lemma 12, and (37) again, Hence, by the inequality above, Corollary 9 and Kronecker’s lemma, we have In order to prove (41), it suffices to prove that Notice thatfor each, we have It follows from Lemma 12 and (37) that, By Kronecker’s lemma, we can get (44) immediately. The proof is complete.

3. Strong Stability for Weighted Sums of AANA Sequence

Theorem 20. Letbe a sequence of AANA random variables with, which is stochastically dominated by a random variable. Letandbe two sequences of positive numbers withand. Denote, . If the following conditions are satisfied:(i); (ii), for some,then there exist,, such that

Proof. For each, denote By Definition 1 and conditions (i), we can obtain By Borel-Cantelli lemma for any sequence, with probability 1, the sequencesandconverge on the same set and to the same limit. We will prove that., which implies (6) with. According to Lemma 7,is a sequence of AANA random variables with mean zero. Byinequality and Lemma 12, we have Notice that where the last inequality follows from the fact that for By (50), (51) and condition (i), (ii), we can get that Therefore, it follows from (53) and Corollary 15 that The proof is complete.

Corollary 21. Suppose that the conditions of Theorem 20 are satisfied andfor each. If, then.

Proof. According to the proof of Theorem 20, we need only to prove that Notice thatfor each; then By Kronecker’s lemma, we can get (55) immediately.

Theorem 22. Letbe a sequence of AANA random variables with mean zero and, which is stochastically dominated by a random variable. Letandbe two sequences of positive numbers withand. Denote,. If the following conditions are satisfied:(i); (ii); (iii), for some,then

Proof. By (49), condition (i), and Borel-Cantelli lemma, it suffices to prove,So we need only to prove We can get (60) from the proof of Corollary 21. In the following, we prove (59). Putandfor. According to Lemma 7,is a sequence of AANA random variables with mean zero. Byinequality and Lemma 12, It is easy to see that Therefore, follows from condition (i), (61) and (62). By Corollary 15 and (63), we can obtain (59) immediately. The proof is complete.

Theorem 23. Letbe a sequence of AANA random variables with, which is stochastically dominated by a random variable. Letandbe two sequences of positive numbers withand. Define,,. If(i)for any;(ii);(iii),then there exist,, such that

Proof. According to Lemma 7,andare sequences of AANA random variables.
Sinceis nondecreasing, then for any which implies that. Therefore, By Borel-Cantelli lemma for any sequence, with probability 1, the sequencesandconverge on the same set and to the same limit. We will prove that., which implies the theorem with. Byinequality and Lemma 12, Since, following from (65) and condition (ii), then we have. To prove, we need to prove thatand: Sinceis nondecreasing andis nonincreasing, then By (67)–(71), we can get that Therefore, it follows from Corollary 9 and Kronecker’s lemma that Taking,, we can get (64). The proof is complete.

Remark 24. Since NA implies AANA, Theorem 20 extends corresponding result for NA random variable (see Wang et al. [24]) to AANA random variables without adding any extra condition.
Similar to the proof of Corollary 21, we can get the following corollary.

Corollary 25. Let the conditions of Theorem 23 be satisfied andfor each. If, then.

Corollary 26. Letbe a sequence of AANA random variables with, which is stochastically dominated by a random variable. Letandbe two sequences of positive numbers withand. Let, whereis a slowly varying function as,. Define,. If(i)for each;(ii).Then there exist,, such that

Proof. It is easy to verify that conditions (i)–(iii) of Theorem 23 hold under the conditions of Corollary 25. So Corollary 25 is true by Theorem 23.

Corollary 27. Suppose that the conditions of Corollary 26 are satisfied. Iffor each, then.

Proof. According to Corollary 26, we need only to prove (60). Byfor eachand Lemma 12, we have Since, we can takesuch that. By Lemma 13 and differential mean value theorem, we can obtain It is easily seen thatis a quasimonotonically increasing function by Lemma 13. Hence, we have by (75) and (76) that By Kronecker’s lemma, we can get (60) immediately. The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are most grateful to the Editor Jaume Giné and anonymous referees for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper. This work is supported by the National Natural Science Foundation of China (11171001, 11201001), the Natural Science Foundation of Anhui Province (1208085QA03, 1308085QA03), Doctoral Research Start-Up Funds Projects of Anhui University, Applied Teaching Model Curriculum of Anhui University (XJYYXKC04), the Students Innovative Training Project of Anhui University (201310357004, cxcy2012003), and the Students Science Research Training Program of Anhui University (KYXL2012007, kyxl2013003).

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