On the Divergence of a Certain Random Series

Abstract

The divergence of the stochastic series $\sum^\infty_{n=1} S_n^+/n$ is investigated, where $S_n^+$ is the positive part of the sum of the first $n$ components of a sequence of independent, identically distributed random variables $\{X_i, i = 1,2, \cdots\}$. It is shown that if $P(X_1 = 0) \neq 1$ then either this series or the companion series $\sum^\infty_{n=1} S_n^-/n$ diverges almost surely. If $EX_1^2 < \infty$ and $EX_1 = 0$ then necessarily both of these series diverge. The method of proof also yields the almost sure divergence of $\sum^\infty_{n=1} S_n/n$. These results are extended to the series $\sum^\infty_{n=1} S_n^+/n^{1+p}$ for $0 \leqq p < \frac{1}{2}$ by a slightly different method of proof which does not, however, yield the divergence of $\sum^\infty_{n=1} S_n/n^{1+p}$.