A theorem about the oscillation of sums of independent random variables

Hanson, D. L.; Wright, F. T.

doi:10.1090/S0002-9939-1973-0315779-0

A theorem about the oscillation of sums of independent random variables
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by D. L. Hanson and F. T. Wright

Proc. Amer. Math. Soc. 37 (1973), 226-233

DOI: https://doi.org/10.1090/S0002-9939-1973-0315779-0

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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 $|z| = 1$ 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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