在条件lim(n→∞)(infP(X_n= 0)>0)下的一类独立随机变量和的收敛定理

Theorems on Convergence of Sums of Independent Random Variables under the Condition lim(n→∞)(infP(X_n= 0)>0)

设｛Ｘ_ｎ，ｎ≥ １｝是一独立随机变量序列．受概率数论中Ｅｒｄｏｓ猜想的启发，我们研究了在条件lim(n→∞)(infP(X_n= 0)>0)下的独立项级数ｓｕｍ ｆｒｏｍ ｎ＝１ Ｘ_ｎ的 ａ．ｓ．收敛性，并且获得了该级数ａ．ｓ．收敛的两个充分必要条件和一个充分条件．这些定理分别改进了文献［３］、［５］中关于Ｅｒｄｏｓ猜想的研究结果．

Let {X_n, n ≥ 1} be a sequence of independent random variables. Motivated by a conjecture of Erdos in probabilistic number theory, we assume n→∞liminP(X_n = 0) > 0 and investigate the a.s. convergence of sum for n=1 X_n. In this paper, we obtain two 'sufficient and necessary' conditions and one 'sufficient' condition of the a.s. convergence of sum for n=1 X_n. In particular, we have improved the related results in [3]. [5].