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On the complete convergence for arrays of rowwise ψ-mixing random variables
Journal of Inequalities and Applications volume 2013, Article number: 393 (2013)
Some sufficient conditions for complete convergence for maximal weighted sums and weighted sums are presented, where is an array of rowwise ψ-mixing random variables, and is an array of constants. The obtained results extend and improve the corresponding result in the previous literature.
The following notion was given firstly by Hsu and Robbins .
Definition 1.1 A sequence of random variables is said to converge completely to a constant θ if for any ,
In this case, we write completely. In view of the Borel-Cantelli lemma, the result above implies that almost surely. Therefore, the complete convergence is a very important tool in establishing almost sure convergence of summation of random variables as well as weighted sums of random variables.
Let be a sequence of random variables, defined on a probability space , and denote σ-algebras
As usual, for a σ-algebra ℱ, we denote by the class of all ℱ-measurable random variables with the finite second moment. Given σ-algebras , ℬ in ℱ, let
Define the mixing coefficients by
Definition 1.2 A sequence of random variables is said to be a ψ-mixing (φ-mixing) sequence of random variables if () as .
Clearly, from the definition above, we know that the independence implies ψ-mixture and φ-mixture. It is easily seen that the ψ-mixing condition is stronger than the φ-mixing. Therefore, the family of ψ-mixing is a special case of φ-mixing. Years after the appearance of Dobrushin , many works of investigation concerning the convergence properties of φ-mixing random variables have emerged. We refer the reader to Ibragimov , Cogburn , Sen , Choi and Sung , Utev , Chen , Shao , Rüdiger , Chen et al. , Zhou , Wang et al. [14, 15], Guo .
However, according to our knowledge, few papers discuss the subjects for sequences or arrays of ψ-mixing random variables except Blum et al. , Bradley , Yang , Wu and Zhu , Wang et al. [14, 15], and Yang and Liu . The goal of this paper is to study a complete convergence for arrays of rowwise ψ-mixing random variables.
Then we recall that the following concept of stochastic domination is a slight generalization of identical distribution.
Definition 1.3 An array of rowwise random variables is said to be stochastically dominated by a nonnegative random variable X (write ) if there exists a constant such that
Stochastic dominance of by the random variable X implies that if the p-moment of X exists, i.e., if .
Hu et al.  obtained the following result in the complete convergence.
Theorem A Let be an array of rowwise independent random variables with . Suppose that are uniformly bounded by some random variable X. If for some , then
The main purpose of this article is to discuss the complete convergence for weighted sums of ψ-mixing random variables. We shall extend Theorem A by considering ψ-mixing instead of independent. It is worthy to point out that our main methods differ from those used by Hu et al. .
Below, C will be used to denote various positive constants, whose value may vary from one application to another. For a finite set A, the symbol denotes the number of elements in the set A. will indicate the indicator function of A.
2 Main results and some lemmas
Now, we state our main results. The proofs will be given in Section 3.
Theorem 2.1 Let be an array of rowwise ψ-mixing random variables with and . Suppose that and for some . Let be a real numbers array satisfying for some . Furthermore, when , we suppose that there exists a constant such that . Then
Take and in Theorem 2.1, we can have the following corollary.
Corollary 2.1 Let be an array of rowwise ψ-mixing random variables with and . Suppose that and for some , then
Remark 2.1 Since the independence implies ψ-mixture, Theorem 2.1 and Corollary 2.1 hold for arrays of rowwise independent random variables. Therefore, Theorem 2.1 and Corollary 2.1 extend and improve Theorem A.
Theorem 2.2 Let be an array of rowwise ψ-mixing random variables with . Suppose that and for some . Let be a real numbers array satisfying for some . Suppose that the following statements hold.
There exists a positive constant such that ;
if . Then
Remark 2.2 Compared with Theorem 2.1, Theorem 2.2 requires a stronger mixing rate, but weakens the requirement of . In fact, holds if , .
Now, we state some lemmas which will be used in the proofs of our main results.
Let be a sequence of ψ-mixing random variables satisfying , . Assume that and for each . Then there exists a constant C depending only on q and such that
Lemma 2.2 (Yang )
Let be a sequence of ψ-mixing random variables with , a.s., , , . Then ,
where , , , .
Lemma 2.3 Let be a sequence of ψ-mixing random variables, and let , , , then
Proof By the definition of ψ-mixing, we have
The proof is complete. □
In this section, we state the proofs of our main results.
Proof of Theorem 2.1 Let , . Since , without loss of generality, we may assume that . Let , where N is a positive integer with . Let
Firstly, we prove . By , we know that . If , we have
If , by and , we also have
Therefore, we know that (1) holds for . Let . To prove , it suffices to show that .
If , by Markov’s inequality and Lemma 2.1, we have
If , take . By and Lemma 2.1, we have
By a similar argument as in the proof of (2) (replacing exponent 2 into q), we can get
Note that and the definition of q, we have
From (3)-(5), we know that (2) still holds for . By (1) and (2), we have
Secondly, we prove . Let and , then
By , we know . Note , we have . Hence, we have , then
Take , then . From , we have
Therefore, by (7) and (8), we have
By the definition of v, we have , then
From (9), (10) and , then
Finally, we prove . Obviously, we know that . Let . We must let M be at least N such that . Take , we have
By Lemma 2.3, we have
From (11) and (12), we have
Noting that . We have
The proof is completed. □
Proof of Theorem 2.2 Following the notations of , and , but let
Obviously, by following the methods used in the proof of (1), we have
By similar arguments as in the proofs of
we can prove
Here, we omit the details. Therefore, we need only to show
Take , by and , we know . Hence, from condition (i), we have
where . Therefore, .
Take . Clearly, when n is sufficiently large. Note that , then when n is sufficiently large. By Lemma 2.2, we have
where . Note that holds if . Therefore, by condition (ii), we have
Hence, when n is sufficiently large, by (14) and (15), we have
The proof is completed. □
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The authors are grateful to the referees for carefully reading the manuscript and for providing some comments and suggestions, which led to improvements in the paper. The research of Yong-Feng Wu was supported by the Humanities and Social Sciences Foundation for the Youth Scholars of Ministry of Education of China (12YJCZH217) and the Natural Science Foundation of Anhui Province (1308085MA03, 1208085MG121). The research of Hui Ding was supported by the NSF of Education Ministry of Anhui province (KJ2012Z278) and the National Statistics Science Research Project (2012LY153).
The authors declare that they have no competing interests.
YFW carried out the proofs of the main results in the manuscript. HD participated in the design of the study and drafted the manuscript. All authors read and approved the final manuscript.
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Wu, YF., Ding, H. On the complete convergence for arrays of rowwise ψ-mixing random variables. J Inequal Appl 2013, 393 (2013). https://doi.org/10.1186/1029-242X-2013-393
- ψ-mixing random variable
- weighted sums
- complete convergence