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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 562838, 8 pages
http://dx.doi.org/10.1155/2012/562838
Research Article

Strong Convergence Properties for Asymptotically Almost Negatively Associated Sequence

1School of Mathematics and Computational Science, Anqing Teachers College, Anqing 246133, China
2College of Water Conservancy and Hydropower Engineering, HoHai University, Nanjing 210098, China
3College of Mathematics and Computation Science, Anhui Normal University, Wuhu 241000, China

Received 22 June 2012; Accepted 10 September 2012

Academic Editor: Garyfalos Papaschinopoulos

Copyright © 2012 Xueping Hu 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

By applying the moment inequality for asymptotically almost negatively associated (in short AANA) random sequence and truncated method, we get the three series theorems for AANA random variables. Moreover, a strong convergence property for the partial sums of AANA random sequence is obtained. In addition, we also study strong convergence property for weighted sums of AANA random sequence.

1. Introduction

A finite family of random variables is said to be negatively associated (in short NA) if for every pair of disjoint subsets of whenever are coordinate-wise nondecreasing such that the covariance exists. An infinite sequence of random variables is said to be NA if every finite subfamily is NA.

The notion of NA was first introduced by Block et al. (1982) [1]. Joag-Dev and Proschan (1983) [2] showed that many well-known multivariate distributions possess the NA property. By inspecting the proof of maximal inequality for NA random variables in Matuła [3], Chandra and Ghosal discovered that one can also allow negative correlations provided they are small. Primarily motivated by this, Chandra and Ghosal [4, 5] introduced the following dependence.

Definition 1.1. A sequence of random variables is said to be asymptotically almost negatively associated, if there exists a nonnegative sequence as such that for all and for all coordinatewise nondecreasing continuous functions and whenever the variances exit.
Obviously, the family of AANA sequences contain NA (in particular, independent) sequences (with ) and some more sequences of random variables which are not much deviated from being NA. An example of an AANA sequence which is not NA was introduced by Chandra and Ghosal [4].
Since the notion of AANA sequence was introduced by Chandra and Ghosal [4], the AANA properties have aroused wide interest because of numerous applications in reliability theory, percolation theory, and multivariate statistical analysis. In the past decades, a lot of effort was dedicated to proving the limit theorems of AANA random variables; we can refer to [410]. Hence, extending the limit properties of AANA random variables has very important significance in the theory and application.
In this paper, we mainly study the strong convergence property for the partial sums of AANA random variables; furthermore the strong convergence property for weighted sums of AANA random variables is also obtained.
Throughout the paper, let be the indicator function of the set , and let for some . The denotes that there exits a positive constant such that . The symbol represents a positive constant which may be different in various places. The main results of this paper are dependent on the following lemmas.

Lemma 1.2 (Yuan and An [6]). Let be a sequence of AANA random variables with mixing coefficients , and let be all nondecreasing (or nonincreasing) functions; then is still a sequence of AANA random variables with mixing coefficients .

Lemma 1.3 (Wang et al. [7]). For , let be a sequence of AANA random variables with mixing coefficients and for each . If , then there exists a positive constant depending only on such that for all where , and is the dual number of .

Lemma 1.4 (Wu [11]). Let be a sequence of random variables. For each , there exists a random variable such that then, for any , the following two statements hold:

Lemma 1.5 (Sung [12]). Let be a positive increasing function on satisfying as , and let be the inverse function of . If and satisfy, respectively, then

2. Strong Convergence for the Partial Sums of AANA Random Variables

Theorem 2.1. Let be a sequence of AANA random variables with , if the following assumptions holds: then almost surely convergence.

Remark 2.2. The proof of Theorem 2.1 is similar to the proof of Theorem   in [11], and by Lemmas 1.2 and 1.3, we omit it.

Theorem 2.3. Let be a sequence of AANA random variables with .
Assume that is a sequence of even functions in , for each , is a positive nondecreasing function in and satisfies one of the following conditions:(i)for there exists a constant such that ;(ii)for , there exists a constant and such that ; however, for , furthermore assume that , for each .
Let be a constant sequence satisfying such that then almost surely convergence, and further it follows from the “Kronecker lemma” that

Proof. For each , denote .
By Lemma 1.2, we can see that, for fixed , is still a sequence of AANA random variables. To verity the Theorem 2.3, for we only need to prove the convergence of three series of (2.1) under condition (i) or (ii). The proof of Theorem 2.3 includes the following three steps.
(1) We prove under condition (i) or (ii).
For each , if satisfies condition (i), noting that is a positive nondecreasing even function in , it is obvious that By (2.2), we can get If satisfies condition (ii), it is easy to prove that (2.5) also holds when .
(2) Next we will show .
If satisfies condition (i), it follows that On the other hand, if condition (ii) holds, according to , for each , we have Hence, it follows from (2.2) that
(3) Finally we prove .
If satisfies condition (i), for each , it is easy to show that by the inequality If condition (ii) holds, according to the inequality, for each , we get Therefore, it also follows from (2.2) that The proof of the Theorem 2.3 is completed by (2.5), (2.8), and (2.11).

Corollary 2.4. Let be a sequence of AANA random variables with , and let be a constant sequence satisfying . For , let , and if satisfies (2.2), then a.s., as .

Proof. It is easy to check that is a sequence of even functions in , for each , is a positive nondecreasing function in, and the following condition holds:

3. Strong Convergence for the Weighted Sums of AANA Random Variables

Theorem 3.1. Let be a different distribution sequence of AANA random variables with and , for each . There exists a random variable satisfying , such that Assume that the following conditions hold for the constant arrays .
(i) ; (ii) for some constant , , where satisfy Lemma 1.5; then

Proof. Let : It suffices to prove that a.s., as . We will estimate each of these terms separately.
To verity a.s., as , we can get from (3.1) and that Hence, by the Borel-Cantelli Lemma it is obvious that a.s., as .
Next we will show that as almost surely. For any , note that , and it follows from the Markov inequality, Lemma 1.2, Lemma 1.3, inequality, and Lemma 1.5 that the last series converges using condition (ii), and by Borel-Cantelli lemma we get a.s., as .
Finally we will prove that a.s., as . Note that ; for each , it is easy to show that by Lemma 1.5, Lemma 1.4, and theKronecker lemma The proof of Theorem 3.1 is completed.

Acknowledgments

This paper is supported by the National Natural Science Foundantion of China (10901003) and the Natural Science Foundation of Anhui Province (KJ2012ZD001, KJ2013A126, KJ2012Z233).

References

  1. H. W. Block, T. H. Savits, and M. Shaked, “Some concepts of negative dependence,” The Annals of Probability, vol. 10, no. 3, pp. 765–772, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. K. Joag-Dev and F. Proschan, “Negative association of random variables, with applications,” The Annals of Statistics, vol. 11, no. 1, pp. 286–295, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. P. Matuła, “A note on the almost sure convergence of sums of negatively dependent random variables,” Statistics & Probability Letters, vol. 15, no. 3, pp. 209–213, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. 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
  5. 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
  6. D. Yuan and J. An, “Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications,” Science in China A, vol. 52, no. 9, pp. 1887–1904, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. 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
  8. 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
  9. 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
  10. J. Baek II, “Almost sure convergence for asymptotically almost negatively associated random variables sequence,” Communications of the Korean Statistical Society, vol. 16, no. 6, pp. 1013–1022, 2009. View at Publisher · View at Google Scholar
  11. Q. Y. Wu, “Probability limit theory for mixing sequence,” Sciences Press, 2005 (Chinese).
  12. S. H. Sung, “Strong laws for weighted sums of i.i.d. random variables. II,” Bulletin of the Korean Mathematical Society, vol. 39, no. 4, pp. 607–615, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH