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 $\sum^\infty_{n=1} S_n/n^{1+p}$.
Citation
L. H. Koopmans. N. Martin. P. K. Pathak. C. Qualls. "On the Divergence of a Certain Random Series." Ann. Probab. 2 (3) 546 - 550, June, 1974. https://doi.org/10.1214/aop/1176996674
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