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June, 1974 On the Divergence of a Certain Random Series
L. H. Koopmans, N. Martin, P. K. Pathak, C. Qualls
Ann. Probab. 2(3): 546-550 (June, 1974). DOI: 10.1214/aop/1176996674


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


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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.


Published: June, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0285.60037
MathSciNet: MR356218
Digital Object Identifier: 10.1214/aop/1176996674

Primary: 60G50
Secondary: 60F05 , 60F20

Keywords: divergence , positive part , Sums of independent identically distributed random , trichtomy

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 3 • June, 1974
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