Abstract and Applied Analysis

Volume 2013 (2013), Article ID 216236, 7 pages

http://dx.doi.org/10.1155/2013/216236

Research Article

On Strong Convergence for Weighted Sums of a Class of Random Variables

Aiting Shen

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

Received 5 February 2013; Accepted 27 February 2013

Academic Editor: Simeon Reich

Copyright © 2013 Aiting Shen. 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.

Let be a sequence of random variables satisfying the Rosenthal-type maximal inequality. Complete convergence is studied for linear statistics that are weighted sums of identically distributed random variables under a suitable moment condition. As an application, the Marcinkiewicz-Zygmund-type strong law of large numbers is obtained. Our result generalizes the corresponding one of Zhou et al. (2011) and improves the corresponding one of Wang et al. (2011, 2012).

1. Introduction

Throughout the paper, let be the indicator function of the set . denotes a positive constant which may be different in various places, and stands for . Denote .

Let be a sequence of identically distributed random variables and an array of constants. The strong convergence results for weighted sums have been studied by many authors; see, for example, Choi and Sung [1], Cuzick [2], Wu [3], Bai and Cheng [4], Chen and Gan [5], Cai [6], Sung [7, 8], Shen [9], Wang et al. [10–14], Zhou et al. [15], Wu [16–18], Xu and Tang [19], and so forth. Many useful linear statistics are these weighted sums. Examples include least squares estimators, nonparametric regression function estimators, and jackknife estimates among others. Bai and Cheng [4] proved the strong law of large numbers for weighted sums:

when is a sequence of independent and identically distributed random variables with and for some and , and is an array of constants satisfying

for some , where .

Cai [6] generalized the result of Bai and Cheng [4] to the case of negatively associated (NA, in short) random variables and obtained the following complete convergence result for weighted sums of identically distributed NA random variables.

Theorem 1. Let be a sequence of NA random variables with identical distributions. And let be a triangular array of constants satisfying for . Let for some . Furthermore, assume that when . If for some , then

Recently, Wang et al. [14] extended the result of Cai [6] for sequences of NA random variables to the case of arrays of rowwise negatively orthant-dependent (NOD, in short) random variables. Sung [8] improved the result of Cai [6] for NA random variables under much weaker conditions. Zhou et al. [15] generalized the result of Sung [8] to the case of -mixing random variables when . The technique used in Sung [8] is the result of Chen et al. [20] for NA random variables, which is not proved for -mixing random variables. The main purpose of the paper is to further study the strong convergence for a class of random variables satisfying the Rosenthal-type maximal inequality by using a different method from that of Sung [8]. We not only generalize the result of Zhou et al. [15] for -mixing random variables to the case of a sequence of random variables satisfying the Rosenthal-type maximal inequality, but also consider the case of . In addition, our main result improves the corresponding one of Wang et al. [11, 14], since the exponential moment condition is weakened to moment condition.

2. Main Results

In this section, we will study the strong convergence for a class of random variables satisfying the Rosenthal-type maximal inequality by using a different method from that of Sung [8]. As an application, the Marcinkiewicz-Zygmund-type strong law of large numbers is obtained.

Theorem 2. Let be a sequence of identically distributed random variables. Let be an array of constants satisfying for some . when . Let for some . Assume that for any , there exists a positive constant depending only on such that

where or . Furthermore, suppose that

for . If

then (3) holds.

Proof. We only need to prove that (3) holds for . The proof for is analogous.

Without loss of generality, we may assume that . It is easy to check that for any ,

which implies that

Firstly, we will show that

When , we have by , Markov’s inequality and (6) that

When , we have by Markov’s inequality and (6) again that

By (10) and (11), we can get (9) immediately. Hence, for large enough,

To prove (3), we only need to show that

Firstly, we will prove (13). By and (6), we can get that

which implies (13).

In the following, we will prove (14). Let . By Markov’s inequality and condition (4), we have

To prove (14), it suffices to show that and .

For and , denote

In view of , it is easy to see that are disjoint and . Hence, we have for all that

which implies that for all ,

By ’s inequality, (13) and (17), we can get that

If , we have by (19) and that

If , we have by (6) and (19) that

By (21) and (22), we can get that . Next, we will prove that .

It follows by (6) and (19) again that

By and , we can get that .

To prove (14), it suffices to show that . By ’s inequality, conditions (5) and (6), we can get that

Therefore, (14) follows from (16) and , immediately. This completes the proof of the theorem.

The following result provides the Marcinkiewicz-Zygmund-type strong law of large numbers for weighted sums of a class of random variables satisfying the Rosenthal-type maximal inequality.

Theorem 3. Let be a sequence of identically distributed random variables. Let be a sequence of constants satisfying for some . when . Let for some . Assume that for any , there exists a positive constant depending only on such that (4) holds, where or . Furthermore, suppose that

for . If (6) holds, then

Proof. The proof of (26) is the same as that of Theorem 2. So the details are omitted. It suffices to show (27). Denote for each . It follows by (26) that

By Borel-Cantelli lemma, we obtain that

For all positive integers , there exists a positive integer such that . We have by (29) that

which implies (27). This completes the proof of the theorem.

If the Rosenthal type inequality for the maximal partial sum is replaced by the partial sum, then we can get the following complete convergence result for a class of random variables. The proof is similar to that of Theorem 2. So the details are omitted.

Theorem 4. Let be a sequence of identically distributed random variables. Let be an array of constants satisfying for some . when . Let for some . Assume that for any , there exists a positive constant depending only on such that

where or . Furthermore, suppose that (5) holds for . If (6) satisfies, then

Remark 5. There are many sequences of dependent random variables satisfying (4) for all . Examples include sequences of NA random variables (see Shao [21]), -mixing random variables (see Utev and Peligrad [22]), -mixing random variables with the mixing coefficients satisfying certain conditions (see Wang et al. [23]), -mixing random variables with the mixing coefficients satisfying certain conditions (see Wang and Lu [24]), and asymptotically almost negatively associated random variables (see Yuan and An [25]). There are also many sequences of dependent random variables satisfying (31) for all . Examples not only include the sequences of above, but also include sequences of NOD random variables (see Asadian et al. [26]) and extended negatively dependent random variables (see Shen [27]).

Acknowledgments

The authors are most grateful to the Editor Simeon Reich and an anonymous referee for careful reading of the manuscript 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 (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 (1308085QA03, 11040606M12, 1208085QA03), the 211 project of Anhui University, the Youth Science Research Fund of Anhui University, and the Students Science Research Training Program of Anhui University (KYXL2012007).

References

B. D. Choi and S. H. Sung, “Almost sure convergence theorems of weighted sums of random variables,” Stochastic Analysis and Applications, vol. 5, no. 4, pp. 365–377, 1987.

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.

W. B. Wu, “On the strong convergence of a weighted sum,” Statistics & Probability Letters, vol. 44, no. 1, pp. 19–22, 1999.

Z. D. Bai and P. E. Cheng, “Marcinkiewicz strong laws for linear statistics,” Statistics & Probability Letters, vol. 46, no. 2, pp. 105–112, 2000.

P. Chen and S. Gan, “Limiting behavior of weighted sums of i.i.d. random variables,” Statistics & Probability Letters, vol. 77, no. 16, pp. 1589–1599, 2007.

G.-h. Cai, “Strong laws for weighted sums of NA random variables,” Metrika, vol. 68, no. 3, pp. 323–331, 2008.

S. H. Sung, “Strong laws for weighted sums of i.i.d. random variables,” Statistics & Probability Letters, vol. 52, no. 4, pp. 413–419, 2001.

S. H. Sung, “On the strong convergence for weighted sums of random variables,” Statistical Papers, vol. 52, no. 2, pp. 447–454, 2011.

A. Shen, “Some strong limit theorems for arrays of rowwise negatively orthant-dependent random variables,” Journal of Inequalities and Applications, vol. 2011, article 93, 2011.

X. Wang, S. Hu, and A. I. Volodin, “Strong limit theorems for weighted sums of NOD sequence and exponential inequalities,” Bulletin of the Korean Mathematical Society, vol. 48, no. 5, pp. 923–938, 2011.

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.

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, vol. 2012, Article ID 315138, 15 pages, 2012.

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.

X. Wang, S. Hu, and W. Yang, “Complete convergence for arrays of rowwise negatively orthant dependent random variables,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, vol. 106, no. 2, pp. 235–245, 2012.

X.-C. Zhou, C.-C. Tan, and J.-G. Lin, “On the strong laws for weighted sums of -mixing random variables,” Journal of Inequalities and Applications, vol. 2011, Article ID 157816, 8 pages, 2011.

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.

Q. Wu, “Sufficient and necessary conditions of complete convergence for weighted sums of PNQD random variables,” Journal of Applied Mathematics, vol. 2012, Article ID 104390, 10 pages, 2012.

Q. Wu, “A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables,” Journal of Inequalities and Applications, vol. 2012, article 50, 2012.

H. Xu and L. Tang, “Some convergence properties for weighted sums of pairwise NQD sequences,” Journal of Inequalities and Applications, vol. 2012, article 255, 2012.

P. Chen, T.-C. Hu, X. Liu, and A. Volodin, “On complete convergence for arrays of rowwise negatively associated random variables,” Theory of Probability and its Applications, vol. 52, no. 2, pp. 393–397, 2007.

Q.-M. Shao, “A comparison theorem on moment inequalities between negatively associated and independent random variables,” Journal of Theoretical Probability, vol. 13, no. 2, pp. 343–356, 2000.

S. Utev and M. Peligrad, “Maximal inequalities and an invariance principle for a class of weakly dependent random variables,” Journal of Theoretical Probability, vol. 16, no. 1, pp. 101–115, 2003.

X. Wang, S. Hu, W. Yang, and Y. Shen, “On complete convergence for weighed sums of -mixing random variables,” Journal of Inequalities and Applications, vol. 2010, Article ID 372390, 13 pages, 2010.

J. F. Wang and F. B. Lu, “Inequalities of maximum of partial sums and weak convergence for a class of weak dependent random variables,” Acta Mathematica Sinica (English Series), vol. 22, no. 3, pp. 693–700, 2006.

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.

N. Asadian, V. Fakoor, and A. Bozorgnia, “Rosenthal's type inequalities for negatively orthant dependent random variables,” Journal of the Iranian Statistical Society, vol. 5, no. 1-2, pp. 66–75, 2006.

A. Shen, “Probability inequalities for END sequence and their applications,” Journal of Inequalities and Applications, vol. 2011, article 98, 2011.