Comparisons between tail probabilities of sums of independent symmetric random variables

We show how estimates for the tail probabilities of sums of independent identically distributed random variables can be used to estimate the tail probabilities of sums of non-identically distributed independent symmetric random variables which are majorized by a single distribution in the sense of Gut's (1992) weak mean domination. As an application, we prove a weak one-sided extension of a law of large numbers of Chen (1978) to a non-identically distributed case and show how some of Gut's (1992) extensions of Hsu-Robbins type laws of large numbers follow from previously known identically distributed cases. We also extend some theorems of Klesov (1993) to the case of weak mean domination. One intermediate result of independent interest is that if X-1, ... ,X-n and Y-1, ... ,Y-n are two collections of independent symmetric random variables such that P(\X-k\ greater than or equal to lambda) less than or equal to P(\Y-k\ greater than or equal to lambda) for every lambda and k, then P(\Y-1 + ... + Y-n\ greater than or equal to lambda) less than or equal to 2P(\X-1+ ... + X-n\ greater than or equal to lambda) for all lambda.