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Some Strong Limit Theorems for Weighted Product Sums of Mixing Sequences of Random Variables
Journal of Inequalities and Applications volumeÂ 2009, ArticleÂ number:Â 174768 (2009)
Abstract
We study almost sure convergence for mixing sequences of random variables. Many of the previous results are our special cases. For example, the authors extend and improve the corresponding results of Chen et al. (1996) and Wu and Jiang (2008). We extend the classical Jamison convergence theorem and the Marcinkiewicz strong law of large numbers for independent sequences of random variables to mixing sequences of random variables without necessarily adding any extra conditions.
1. Introduction and Lemmas
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 , and write . Given algebras in , let
where . Define the mixing coefficients by
Obviously and except in the trivial case where all of the random variables are degenerate.
Definition 1.1.
A random variables sequence is said to be a mixing random variables sequence if there exists such that .
mixing is similar to mixing, but both are quite different. A number of writers have studied mixing random variables sequences and a series of useful results have been established. We refer to Bradley [1] (which assumes in the central limit theorem), Bryc and SmoleÅ„ski [2], Goldie and Greenwood [3] (which assumes ), and Yang [4] for moment inequalities and the strong law of large numbers, Wu [5, 6], Wu and Jiang [7], Peligrad and Gut [8], and Gan [9] for almost sure convergence and Utev and Peligrad [10] for maximal inequalities and the invariance principle. When these are compared with the corresponding results of independent random variables sequences, there still remains much to be desired.
Lemma 1.2 (see [7, Theorem ]).
Let be a mixing sequence of random variables which satisfies
Then converges almost surely (a.s.) and in quadratic mean.
Lemma 1.3 (see [11, Lemma ]).
For each positive integer , let denote the set of all vectors such that .
Then for each positive integer , there exists a function such that the following holds.
For any integer and any choice of real numbers , one has that
2. Main Results and the Proof
To state our results, we need some notions. Throughout this paper, let be a sequence of positive real numbers, and let satisfy .
Jamison et al. [12] proved the following result. Suppose that are i.i.d. random variables with . Denote , that is, the number of subscripts such that . If , then a.s. Chen et al. [13] extended the Jamison Theorem and obtained the following result. Suppose that are i.i.d. random variables with for some . If , then a.s.
The main purpose of this paper is to study the strong limit theorems for weighted sums of mixing random variables sequences and try to obtain some new results. We establish weighted partial sums and weighted product sums strong convergence theorems. Our results in this paper extend and improve the corresponding results of Chen et al. [13], Wu and Jiang [7], the classical Jamison convergence theorem, and the Marcinkiewicz strong law of large numbers for independent sequences of random variables to mixing sequences of random variables.
Theorem 2.1.
Let be a sequence of mixing random variables with , and let the following conditions be satisfied:
where Then
Theorem 2.2.
Suppose that the assumptions of Theorem 2.1 hold, and also suppose
Then for all
Corollary 2.3.
Let be a sequence of mixing identically distributed random variables. Let for some
Then (2.6) holds.
Remark 2.4.
Let be i.i.d. random variables, and in Corollary 2.3, then Corollary 2.3 is the wellknown Jamison convergence theorem. Thus, our Theorem 2.2 and Corollary 2.3 generalize and improve the Jamison convergence theorem from the i.i.d. case to mixing sequence. In addition, by Theorems and in Chen et al. [13] are special situation of Corollary 2.3.
Theorem 2.5.
Let be a sequence of mixing random variables. Let be a sequence of positive real numbers with and let the following conditions be satisfied:
Then for all
Corollary 2.6.
Let be a mixing identically distributed random variable sequence, for , and for . Then for all
In particular, taking , the above formula is the wellknown Marcinkiewicz strong law of large numbers. Thus, our Theorem 2.5 and Corollary 2.6 generalize and improve the Marcinkiewicz strong law of large numbers from the i.i.d. case to mixing sequence. In addition, by Theorem in Wu and Jiang [7] is a special case of Corollary 2.6.
Proof of Theorem 2.1.
Let . From (2.2),
By the BorelCantelli lemma and the Toeplitz lemma,
By and (2.1),
By (2.3),
Applying Lemma 1.2,
converges. Hence
from the Kronecker lemma. Combining (2.15)â€“(2.19), (2.4) holds. This completes the proof of Theorem 2.1.
Proof of Theorem 2.2.
By Lemma 1.3,
where denote the set of all vectors , such that , and are constants which do not depend on and Thus, in order to prove (2.6), we only need to prove that
When , by Theorem 2.1, (2.21) holds. When we get
from the elementary inequality valid for applied with . Hence, in order to prove (2.21), we only need to prove that
By (2.3), using Lemma 1.2, we get that
converges. By the Kronecker lemma, Thus,
that is,
By (2.3),
By Lemma 1.2, we have that
converges. By the Kronecker lemma,
By and the Toeplitz lemma,
Then combining (2.5), we obtain
Substituting (2.29) and (2.31) in (2.26), we get
Then combining (2.2) and the BorelCantelli lemma, (2.23) holds. This completes the proof of Theorem 2.2.
Proof of Corollary 2.3.
By Theorem 2.2, we only need to verify (2.1)â€“(2.3) and (2.5). From having identically distribution, (2.7), (2.8) and (2.5) holds automatically.
Since
by the Toeplitz lemma,
That is,(2.1) holds.
By (2.7),
That is, (2.2) holds.
Similarly,
That is, (2.3) holds. This completes proof of Corollary 2.3.
Proof of Theorem 2.5.
Similar to the proof of Theorem 2.2, by Lemma 1.3, in order to prove (2.12), we only need to prove that
Let then
â€‰(i) When , by (2.10) and (2.11), in order to prove we only need to prove that
By (2.9) and Lemma 1.2,
converges. By the Kronecker lemma, (2.39) holds.

(ii)
When , by (2.9) and the Kronecker lemma,
By (2.10) and the BorelCantelli lemma,
Hence, combining (2.39), (2.37) holds. This completes the proof of Theorem 2.5.
Proof of Corollary 2.6.
Let . We can easy to verify (2.9)â€“(2.11). By Theorem 2.5, Corollary 2.6 holds.
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Acknowledgments
The authors are very grateful to the referees and the editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper. This work was supported by the National Natural Science Foundation of China (10661006), the Support Program of the New Century Guangxi China Tenhundredthousand Talents Project (2005214), and the Guangxi, China Science Foundation (0991081).
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Wu, Q., Jiang, Y. Some Strong Limit Theorems for Weighted Product Sums of Mixing Sequences of Random Variables. J Inequal Appl 2009, 174768 (2009). https://doi.org/10.1155/2009/174768
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DOI: https://doi.org/10.1155/2009/174768