Almost Sure Convergence For Random Sequence Under Moment Conditions

In ref.[1], Fazekas and Klesov obtained almost sure convergence for the sums of random sequence by using Hdjek-Renyi type maximal inequality. In ref.[2], the author further studied the convergence rate for the sums of random sequence in ref.[l].In this paper, we mainly study almost sure convergence for the sums of random sequence under two kinds of superadditive properties. We have the following results:Let {X_n , n≥1} be a sequence of random variables having the rth momentfunction of superadditive structure. Set g(0, n) = g_n and let g_n be nondecreasingsequence. There exists a nondecreasing sequence of positive numbers {b_n}, and limb_n =∞. Ifthenand with convergence ratewhere<*l=g?> ?k=gak-glx, k>\.Let a > 1 and r ^ 1 be given real numbers. 1 ^ O < 2{a'])la. There exists a function g(i, j) having Q - superadditive structure such thatSet g(\,n) = an , ao-O and let a,, be nondecreasing sequence. There exists a nondecreasing sequence of positive numbers \bn}, and limin = oo. Ifthenlim—^- = 0, a.s. ,and with convergence ratewhereB =max6ivfr, VO<^<1, v =Y—a, = a,a , ak=a" - a"_, , A: > 1.