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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A theorem about the oscillation of sums of independent random variables


Authors: D. L. Hanson and F. T. Wright
Journal: Proc. Amer. Math. Soc. 37 (1973), 226-233
MSC: Primary 60G50; Secondary 60J15
MathSciNet review: 0315779
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Abstract: Let $ {X_1},{X_2}, \cdots $ be i.i.d. random variables and let $ {S_n} = {X_1} + \cdots + {X_n}$. The relationship between the tth moment of $ {X_1}$ and the convergence of the series $ \sum\nolimits_{n = 1}^\infty {{z^n}{n^{t - 1}}P({S_n} > 0)} $ is investigated in this paper. The convergence of the series above when $ \vert z\vert = 1$ but $ z \ne 1$ is related to the oscillation of the sequence $ \{ P({S_n} > 0)\} $ and to the oscillation of the sequence $ \{ {S_n}\} $ about zero.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0315779-0
PII: S 0002-9939(1973)0315779-0
Keywords: Oscillation, oscillation about zero
Article copyright: © Copyright 1973 American Mathematical Society