### On the Divergence of a Certain Random Series

L. H. Koopmans, N. Martin, P. K. Pathak, and C. Qualls
Source: Ann. Probab. Volume 2, Number 3 (1974), 546-550.

#### 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 $\sum^\infty_{n=1} S_n/n^{1+p}$.

First Page:
Primary Subjects: 60G50
Secondary Subjects: 60F05, 60F20
Full-text: Open access
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1176996674
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.1214/aop/1176996674
Mathematical Reviews number (MathSciNet): MR356218
Zentralblatt MATH identifier: 0285.60037