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Journal of Applied Mathematics
Volume 2014, Article ID 612140, 8 pages
http://dx.doi.org/10.1155/2014/612140
Research Article

Sufficient and Necessary Conditions of Complete Convergence for Weighted Sums of -Mixing Random Variables

1School of Economics and Management, Fuyang Normal College, Fuyang 236037, China
2School of Economics and Management, Beijing University of Posts and Telecommunications, Beijing 100876, China

Received 12 November 2013; Revised 23 January 2014; Accepted 11 February 2014; Published 19 March 2014

Academic Editor: Fernando Simões

Copyright © 2014 Chongfeng Lan. 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 equivalent conditions of complete convergence are established for weighted sums of -mixing random variables with different distributions. Our results extend and improve the Baum and Katz complete convergence theorem. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for sequence of -mixing random variables is obtained.

1. Introduction

Let be a probability space. The random variables we deal with are all defined on . Let be a sequence of random variables. For each nonempty set , write . Given -algebras , in , let where . Define the -mixing coefficients by where (for a given positive integer ) this sup is taken over all pairs of nonempty finite subsets , of such that dist .

Obviously , , and except in the trivial case where all of the random variables are degenerate.

Definition 1. A sequence of random variables is said to be a -mixing sequence of random variables if there exists such that .
Note that if is a sequence of independent random variables, then for all . -mixing is similar to -mixing, but both are quite different. is defined by (2) with index sets restricted to subsets of and subsets of of , , . On the other hand, -mixing sequence assumes the condition , but -mixing sequence assumes the condition that there exists such that ; from this point of view, -mixing is weaker than -mixing.

The concept of -mixing random variables was introduced by Bradley [1] and a number of limit theories for -mixing sequences of random variables have been established by many authors. We refer to Bradley [1] for the central limit theorem, Bryc and Smoleński [2], Peligrad and Gut [3], and Utev and Peligrad [4] for moment inequalities, Gan [5], Kuczmaszewska [6], and Wu and Jiang [7] for almost sure convergence, and Cai [8], Zhu [9], An and Yuan [10], Zhou et al. [11], Shen and Hu [12], Guo and Zhu [13], Wang et al. [14], and Sung [15, 16] for complete convergence.

A sequence of random variables converges completely to the constant if

In view of the Borel-Cantelli lemma, this implies that almost surely. Hence, complete convergence is one of the most important problems in probability theory. Since the concept of complete convergence was introduced by Hsu and Robbins [17], there have been many authors who devoted the study to complete convergence for independent and identically distributed random variables. One of the most important results is Baum and Katz theorem [18]. The theorem was further generalized and extended in different ways. Katz [19] and Chow [20] formed the following generalization with a normalization of Marcinkiewicz-Zygmund type theorem for the strong law of large numbers.

Theorem 2 (see [21]). Let , , and let be a sequence of independent and identically distributed random variables. If , assume that . Then the following statements are equivalent:(i); (ii) for all .

In many stochastic models, the assumption of independence among random variables is not plausible. So it is necessary to extend the concept of independence to dependence cases. Peligrad and Gut [3] extended this result from independent and identically distributed case to the case of -mixing random variables with identical distribution. But they did not prove whether the result of Theorem 2 of the case holds for -mixing sequence. In practical applications it is difficult to check the independence of a sample or the samples are not independent observations. Therefore, in recent investigations limit theorems are very often considered for sequences of dependent random variables. Recently, a number of limit theorems for dependent random variables have been established by many authors. We can refer to Sung [22], Wu and Jiang [23], Wu [24], and Shen [25].

Let be a sequence of identically distributed random variables and let be an array of constants. The strong convergence results for weighted sums have been studied by many authors; see, for example, Cuzick [26], Choi and Sung [27], Bai and Cheng [28], Chen and Gan [29], and so forth. Many useful linear statistics are weighted sums. Examples include least squares estimators, nonparametric regression function estimators, and jackknife estimates.

Inspired by Theorem 2.1 of Kuczmaszewska [30], our main purpose in this work is to extend the complete convergence for weighted sums of independent and identically distributed random variables to the case of -mixing random variables. However, our proven methods are different from the ones by Kuczmaszewska [30]; by applying inequality (13) of Lemma 10 our proof is much simpler than the one by Kuczmaszewska. Our proof of necessary condition (using Lemma 10) is original. We provide sufficient and necessary conditions of complete convergence for weighted sums of -mixing random variables with different distributions. As applications, the Baum and Katz type result and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of -mixing random variables are obtained. In addition, our main results extend and improve the corresponding results of Peligrad and Gut [3].

Throughout this paper, the symbol denotes a positive constant which is not necessarily the same one in each appearance, will mean for sufficiently large , will mean , and is the indicator function of event .

2. Main Results

Now we state our main results of this paper. The proofs will be given in Section 3.

Theorem 3. Let be a random variable and let be a sequence of -mixing random variables satisfying the condition for all , all , and some positive constant . Let be a sequence of real numbers such that where means and . Let , , and if , assume that , . Then the following statements are equivalent: (i), (ii).

Remark 4. When proving the limit theorem of -mixing random variables with different distributions, many authors apply the condition of being stochastically dominated by , that is, for some constant , , for all , , which implies that , but the converse is not true. Hence our condition of (4) is weaker than the condition of stochastic dominance.

When is a sequence of -mixing identically distributed random variables and , for all , , then Theorem 3 becomes Baum and Katz complete convergence theorem as follows.

Corollary 5. Let be a sequence of -mixing identically distributed random variables. Let , , and if , assume that , . Then the following statements are equivalent:(i); (ii) for all .

Remark 6. Corollary 5 not only generalizes Theorem 2 to -mixing case, but also extends Theorem 2 of Peligrad and Gut [3] to the case . Therefore, Corollary 5 improves and extends the well-known Baum and Katz theorem.

An and Yuan [10, Theorem 2] presented a Marcinkiewicz-Zygmund type strong law of large numbers for -mixing sequence. We find that the proof of their Theorem 2 is wrong because the theorem is based on Theorem 1 [10]. However, the author thinks that their proofs of Theorem 1 have a little problem, since condition (1.2) does not hold for the array with . An and Yuan [10, Theorem 1] proved the implication (i) (ii) under condition (1.3) and proved the converse under conditions (1.2) and (1.3). However, the array satisfying both (1.2) and (1.3) does not exist. Noting that , we have that . But this does not hold when is fixed and is large enough. In this paper, we obtain a new complete convergence result for weighted sums of -mixing random variables without assumption of identical distribution. Our result generalizes and sharpens the result of An and Yuan [10]. The following corollary provides the Marcinkiewicz-Zygmund type strong law of large numbers of -mixing random variables without assumption of identical distribution.

Corollary 7. Let be a random variable and let be a sequence of -mixing random variables satisfying the condition for all , all , and some positive constant .   for some and if , assume that , . Then, for any ,

3. Proof of Main Results

The following lemmas are useful for the proof of the main results.

Lemma 8 (see [4]). Suppose is a positive integer, , and . Then there exists a constant such that the following statement holds.
If is a sequence of random variables such that and and for all , then, for every ,

Lemma 9 (see [30]). Let be a sequence of random variables which is weakly mean dominated by a random variable ; that is, for all and some positive constant ,
Then for any , , and , the following three statements hold:

Lemma 10. Let be a sequence of -mixing random variables. Then there exists a positive constant such that, for any and all ,

Proof of Lemma 10. Since , we have
Note that
Obviously, by Lemma 8, we get
Combining with the Cauchy-Schwarz inequality and (16), we obtain
Now, we substitute (17) into (15) and then into (14), which implies that (13) holds.

Consequently, we prove our main results.

Proof of Theorem 3. First, we prove that .
Note that , where and . To prove (ii) it suffices to show that
Thus, without loss of generality, we may assume that for all , .
For fixed , denote that
Firstly, we show that
If , by , (i), (5), (12) of Lemma 9, and , we have
If , ,   by (5), (11) of Lemma 9, Markov inequality, and from (i), we get
If , , by , we can get , and thus
Note that, if , we have
Hence, by Kronecker lemma and (23), we obtain
From (21), (22), and (25) we can get (20) immediately. Hence, for all sufficiently large and any , we have
It is easy to check that for all sufficiently large which implies that for all sufficiently large
Therefore, in order to prove (ii), we only need to prove that
By (4), (5), and , we can get that
That is, (29) holds. Thus, it remains to prove (30).
Since is a sequence of -mixing random variables and , , thus is still a sequence of -mixing random variables with . By Markov inequality and Lemma 8, let . Then,
When , taking , then , and by Jensen inequality and (11) of Lemma 9, we have
Taking , we have by (11) of Lemma 9 that
When , then, taking , by (32), we get
Similarly to the proof of inequality (34), we obtain which implies that
Now, we prove the converse. To prove that (ii) implies (i), it suffices to show that
Noting that then from (ii) and (5), we have
Combining with the condition of ,
Thus, for sufficiently large ,
Therefore, by applying Lemma 10, it is easy to see that which, together with the conditions of (4) and (40), gives which implies that (i) holds. This completes the proof of Theorem 3.

Proof of Corollary 7. Taking and , for all , in Theorem 3, we can get (7) immediately; thus
It follows from Borel-Cantelli lemma that
Hence,
For all positive integers , there exists a nonnegative integer such that . We have by (47) that which implies that
The proof of Corollary 7 is completed.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to the editor and the anonymous referees for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper. The research is supported by the National Natural Science Foundation of China (11226200) and the Natural Science Research Project of Anhui Province (KJ2013Z265).

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