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Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 140548, 10 pages
http://dx.doi.org/10.1155/2008/140548
Research Article

On Chung-Teicher Type Strong Law of Large Numbers for -Mixing Random Variables

Department of Applied Mathematics, Lublin University of Technology, Nadbystrzycka 38 D, 20-618 Lublin, Poland

Received 23 December 2007; Revised 5 March 2008; Accepted 20 March 2008

Academic Editor: Stevo Stevic

Copyright © 2008 Anna Kuczmaszewska. 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

In this paper the classical strong laws of large number of Kolmogorov, Chung, and Teicher for independent random variables were generalized on the case of -mixing sequence. The main result was applied to obtain a Marcinkiewicz SLLN.

1. Introduction

Let be a sequence of random variables defined on the probability space with value in a real space and let . We say that the sequence satisfies the strong law of large numbers (SLLN) if there exists some increasing sequence and some sequence such thatIn this paper, we consider the strong law of large numbers for sequences of dependent random variables which are said to be -mixing. To introduce the concept of -mixing sequence, we need the maximal correlation coefficient defined as follows:where , are the finite subsets of positive integers such that dist and is the -field generated by the random variables .

Definition 1.1. A sequence of random variables is said to be a -mixing sequence if

-mixing random variables were investigated by many authors. Various moment inequalities for sums and maximum of partial sums can be found in papers by Bradley [1], Bryc and Smoleński [2], Peligrad [3], Peligrad and Gut [4], and Utev and Peligrad [5]. These inequalities are used in many papers concerning the problems of invariance principle (Utev and Peligrad [5]), CLT (Peligrad [3]), or complete convergence for some stochastically dominated sequence of -mixing random variables (Cai [6]), and for an array of rowwise -mixing random variables (Zhu [7]). They will be also important in our further consideration.

The aim of this paper is to give some sufficient conditions for SLLN for a sequence of -mixing random variables without assumptions of identical distribution and stochastical domination. The result presented in this paper is obtained by using the maximal type inequality and the following strong law of large numbers proved by Fazekas and Klesov [8].

Theorem 1.2 (Fazekas and Klesov [8]). Let be a nondecreasing, unbounded sequence of positive numbers. Let be nonnegative numbers. Let be a fixed positive number. Assume that for each : Ifthen

Using this theorem, we are going to show that classical Kolmogorov, Chung, and Teicher's strong law of large numbers for independent random variables (Chung [9] and Teicher [10]) can be generalized to the case of -mixing sequences.

In our further consideration, we need the following result.

Lemma 1.3 (Utev and Peligrad [5]). Let be a -mixing sequence with , , , . Then, there exists a positive constant such that

Let denote a constant which is not necessary the same in its each appearance.

2. The Main Result

Let be a sequence of nonnegative, even, continuous and nondecreasing on functions with and such that for all and :

Theorem 2.1. Let be a sequence of -mixing random variables and let be an increasing sequence of positive real numbers. Let .
Assume that satisfies in (2.1) with
(A)(B)
for some sequence of positive numbers such that (C) or satisfies in (2.1) with (A1) and for some sequence of positive numbers such that (C1)
Then,

Proof. Let , , , and .
Then,for in the case (a) or in the case (b). Hence, the sequences and are equivalent in Khinchin's sense.
Thus, we need only to show thatBy Lemma 1.3, for , we haveBy Theorem 1.2 applied withwe see that in order to show (2.9) it is enough to prove thatwhich holds ifPut . Then, we haveMoreover, we note thatHence, in case (a) we get, by (B) and (C),while in case (b) we get, by (B) and (C1),Using the fact thatboth in either case (a) or case (b) Thus, we have established (2.9) and consequently (2.7).

Corollary 2.2. Let be a sequence of -mixing random variables satisfying the condition with for and all , or for .
Then,

Proof. Let for any . Then, the assumption (B) of Theorem 2.1 is fulfilled.
Indeed we see by (2.21) with for or for .
Let now . Then, for , , , the conditions (A) and (C) with are fulfilled: by (2.21) with . by (2.23) and (2.21) with , .
Thus, by Theorem 2.1, we havefor .
Now, we need to show that (2.26) also holds for .
Let . Then, for , by the similar calculations as in case , we get by (2.21) with and by (2.23) and (2.21) with .
Therefore, by Theorem 2.1, we get (2.26) for .
This completes the proof of Corollary 2.2.

Corollary 2.3. Let be a sequence of -mixing random variables with , . Let for all , and some positive constant .
Moreover, when , let . Then,

Proof. We first note that (2.29) impliesfor any and .
Put now for any .
It is easy to see that , (2.29) and (2.31) with imply convergence of the series:for , in the case and in the case .
We haveBecause of Corollary 2.2, this proves that (2.26) holds for any .
To complete the proof we should show thatfor any .
For and , we haveButso by Kronecker's lemma we getMoreover, we noteand by Kronecker's lemmawhich together with (2.35) and (2.37) gives (2.34) for .
Let . First, we will show thatTo achieve this, we put for .
Because of and , we haveTherefore, for large enough , we obtainHence, we need only to show thatBy , we getwhich implies (2.43).
By Lemma 1.3 and (2.23) with ,which gives (2.44).
Thus, we have established that (2.40) holds true. Equations (2.40) and (2.26) imply (2.34) which completes the proof of Corollary 2.3.

Corollary 2.3 gives the Marcinkiewicz SLLN for -mixing random variables stochastically dominated by a random variable for . The identical result was obtained by Cai [6] as a consequence of complete convergence theorem. Both of them, for , are a special case of more general result of Fazekas and Tómács [11] obtained for stochastically dominated random variables without assuming any kind of dependence.

Acknowledgment

The author thanks the referees for their comments and suggestions which allowed to improve this paper.

References

  1. R. C. Bradley, “On the spectral density and asymptotic normality of weakly dependent random fields,” Journal of Theoretical Probability, vol. 5, no. 2, pp. 355–373, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. W. Bryc and W. Smoleński, “Moment conditions for almost sure convergence of weakly correlated random variables,” Proceedings of the American Mathematical Society, vol. 119, no. 2, pp. 629–635, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. Peligrad, “Maximum of partial sums and an invariance principle for a class of weak dependent random variables,” Proceedings of the American Mathematical Society, vol. 126, no. 4, pp. 1181–1189, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Peligrad and A. Gut, “Almost-sure results for a class of dependent random variables,” Journal of Theoretical Probability, vol. 12, no. 1, pp. 87–104, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. G.-H. Cai, “Strong law of large numbers for ρ-mixing sequences with different distributions,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 27648, 7 pages pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M.-H. Zhu, “Strong laws of large numbers for arrays of rowwise ρ-mixing random variables,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 74296, 6 pages pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  8. I. Fazekas and O. Klesov, “A general approach to the strong laws of large numbers,” Theory of Probability and Its Applications, vol. 45, no. 3, pp. 436–449, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. K.-L. Chung, “Note on some strong laws of large numbers,” American Journal of Mathematics, vol. 69, no. 1, pp. 189–192, 1947. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. H. Teicher, “Some new conditions for the strong law,” Proceedings of the National Academy of Sciences of the United States of America, vol. 59, no. 3, pp. 705–707, 1968. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. I. Fazekas and T. Tómács, “Strong laws of large numbers for pairwise independent random variables with multidimensional indices,” Publicationes Mathematicae Debrecen, vol. 53, no. 1-2, pp. 149–161, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet