在条件\liminf _\boldsymboln \rightarrow \infty \mathrmP\left(\boldsymbolX_\boldsymboln=0\right)>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 \liminf _\boldsymboln \rightarrow \infty \mathrmP\left(\boldsymbolX_\boldsymboln=0\right)>0

摘要: 设｛X_n，n≥ 1｝是一独立随机变量序列．受概率数论中Erdös猜想的启发，我们研究了在条件\liminf _\boldsymboln \rightarrow \infty \mathrmP\left(\boldsymbolX_\boldsymboln=0\right)>0下的独立项级数\sum_n=1^\infty 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Erdös猜想的研究结果。

Abstract: Let X_n，n≥ １ be a sequence of independent random variables. Motivated by a conjecture of Erdos in probabilistic number theory, we assume \liminf _\boldsymboln \rightarrow \infty \mathrmP\left(\boldsymbolX_\boldsymboln=0\right)>0 and investigate the a.s. convergence of sum for \sum_n=1^\infty X_n. In this paper, we obtain two "sufficient and necessary" conditions and one "sufficient" condition of the a.s. convergence of sum for \sum_n=1^\infty X_n. In particular, we have improved the related results in 3. 5.