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On the Brunk-Chung type strong law of large numbers forsequences of blockwise m-dependent random variables

Published online by Cambridge University Press:  03 May 2006

Le Van Thanh
Le Van Thanh*
Affiliation:
Department of Mathematics, Vinh University, Vinh, Nghe An 42118, Vietnam. lvthanhvinh@yahoo.com
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Abstract

For a sequence of blockwise m-dependent random variables {Xn,n ≥ 1}, conditions are provided under which $\lim_{n\to\infty}(\sum_{i=1}^nX_i)/b_n=0$ almost surely where {bn,n ≥ 1} is a sequence of positive constants. The results are new even when bn ≡ nr,r > 0. As special case, the Brunk-Chung strong law of large numbers is obtained for sequences of independent random variables. The current work also extends results of Móricz [Proc. Amer. Math. Soc.101 (1987) 709–715], and Gaposhkin [Teor. Veroyatnost. i Primenen. 39 (1994) 804–812]. The sharpness of the results is illustrated by examples.

Keywords

Strong law of large numbersalmost sure convergenceblockwise m-dependent random variables.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 10 , February 2006 , pp. 258 - 268
Copyright
© EDP Sciences, SMAI, 2006

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References

Chobanyan, S., Levental, S. and Mandrekar, V., Prokhorov blocks and strong law of large numbers under rearrangements. J. Theoret. Probab. 17 (2004) 647–672. CrossRef
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V.F. Gaposhkin, On the strong law of large numbers for blockwise independent and blockwise orthogonal random variables. Teor. Veroyatnost. i Primenen. 39 (1994) 804–812 (in Russian). English translation in Theory Probab. Appl. 39 (1994) 667–684 (1995).
M. Loève, Probability Theory I. 4th ed. Springer-Verlag, New York (1977).
Móricz, F., Strong limit theorems for blockwise m-dependent and blockwise quasiorthogonal sequences of random variables. Proc. Amer. Math. Soc. 101 (1987) 709–715. CrossRef