Chúng tôi khái quát một số định lý của Châu Tinh Trì, Lai [2] β chung khoản tiền trọng của IID biến ngẫu nhiên. Một đặc điểm của thời điểm điều kiện như Eeα | Vietnam Journal of Mathematics 33. V Í e It ini ai m J o mt r im ai I of MATHEMATICS VAST 2005 On the Almost Sure Convergence of Weighted Sums of . Random Variables Dao Quang Tuyen Institute of Mathematics 18 Hoang Quoc Viet Road 10307 Hanoi Vietnam Abstract. . I 1 . . 1. Introduction Let 1 2 be independent identically distributed random variables with zero means. Let . 1 any array of real numbers and . be any sequence of positive integers such that . . The problem is to find best conditions for almost sure convergence to zero of i S . . 1 Some convergence theorems for . have been obtained by Chow 1 Chow and Lai 2 Hanson and Koopman 4 Pruitt 5 and Stout 6 . In 2 Chow and Lai have proved strong theorems for the case . . where 0 and . satisfies some summable conditions like limsup . n. or limsup . .In this paper we generalize some of these results to more general . . In addition we give a characterization of general moment condition like 1 by almost sure convergence to zero of . a. f . For example one such known result 2 Theorem 1 states that 1i for any 1 if and only if . 0 . Dao Quang Tuyen 2. Results We shall use the following definition. An array . is said to converge to a sequence . almost uniformly as k if for every 0 there exists such that . - . for all and all k except at most k for each . It is obvious that if . . almost uniformly then . . for all . Note that for arrays uniform convergence implies almost uniform convergence. But the converse is not true. The array in the proof of Corollary 2 is an example. Theorem 1. Let 1 2 be . mean 0 random variables. Then 11 for all 0 if and only if . 1 . . 0 . for every array of real numbers . satisfying a A . 2 for all 1 . b 0 almost uniformly - c TA for some 0. 1 Thịs theorem improves Theorem 2 in 2 which deals with . . - log where 1 k. . This array clearly satisfies a b and c of Theorem 1. Theorem 2. Let . be any sequence of . mean 0 random variables . be any array of real numbers and . be .
