ESAIM: Probability and Statistics (ESAIM: P&S)

ESAIM: P&S, April 2006, Vol. 10, pp. 258-268
DOI: 10.1051/ps:2006010

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

Le Van Thanh

Department of Mathematics, Vinh University, Vinh, Nghe An 42118, Vietnam. lvthanhvinh@yahoo.com

(Received September 5, 2005. / Published online: 3 May 2006)

Abstract
For a sequence of blockwise m-dependent random variables $\{X_n,n\geq 1\}$ , conditions are provided under which $\lim_{n\to\infty}(\sum_{i=1}^nX_i)/b_n=0$ almost surely where $\{b_n,n\geq 1\}$ is a sequence of positive constants. The results are new even when $b_n\equiv n^r,r>0$ . 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.

Mathematics Subject Classification. 60F15.

Key words: Strong law of large numbers, almost sure convergence, blockwise m-dependent random variables.

© EDP Sciences, SMAI 2006