Theorems on Convergence of Sums of Independent Random Variables under the Condition \liminf _\boldsymboln \rightarrow \infty \mathrmP\left(\boldsymbolX_\boldsymboln=0\right)>0

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.