Some New Results For SLLN Of The Sums Of Random Variables Under Moment Restrict

In general,there are two basic approaches to proving the strong law of large numbers(SLLN). The first is to prove the SLLN for a subsequence ofSn/Bn then reduce the problem for the whole sequenceto that of the subsequence. In so doing, a maximal inequality for cumulative sums is usually needed for the second step. The second approach is to use directly a maximal inequality named H(?)jek-R(?)nyi type, referring to the paper by H(?)jek and R(?)nyi [2] devoted to independent summands. Inequalities of this type are not easy to obtain and the first approach prevails. However, after a Hdjek-Renyi type inequality is obtained, the proof of the SLLN becomes an obvious problem.The SLLN is a key question of Probability Theory. And estimates of the rate of convergence in SLLN occupy an inportant place. Strasen[3], Siegmund[4] and James[5] got some rates of convergence throng studing the asymptotic equivalence ofBut these results need some assumptions such as i.i.d. Recently, Fazekas and Klesov [1] obtained a H(?)jek-R(?)nyi type maximal inequality and then proved the SLLN for general summands, which generalized some results about dependent summands. In this paper, the author futher studied the strong convergence rate for sums of random variables, under the same conditions as that in [l],and we got the better results. When the conditions of Theorem 2.1 of [1] are the same as our Theorem 2.1.1, it only gives limSn/bn = 0,a.s. in [1], but our Theorem 2.1.1 gives more accurate results: the a.s.convergence rate for Sn/bn convergence to zero. And Theorem 2.1.1 hasnothing to do with the dependent structures of random variables. The other results in present paper are also more accurate than the ones in [1].As applications, we generalized the SLLN for - mixing and - mixingsequences in [6], and we allow in Theorem 3.2.1.we also gave the SLLN and its rates about martingale difference sequences , linear processes, NA sequences and - mixingale sequences.For martingale difference sequence {Xi} , Chow Y S [10] (or [8]theory3.3.8)proved that lim Sn /n = 0,a.s. when helds, but ourCorollary 3.1.1 gives more accurate results: the a. s. convergence rate O for convergence to zero. Especially, if q = 1, 1in Corollary 3.1.1, then we obtain that limSn/n = 0 a. s. and the convergence rateSn/n = O(n-5/2 ),a.s. 0 < < 1. If we take bn=n in Theorem 3.3.4, We can get0,a.s. for linear process { X ?} and the convergence rate...