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December, 1981 The Growth of Random Walks and Levy Processes
William E. Pruitt
Ann. Probab. 9(6): 948-956 (December, 1981). DOI: 10.1214/aop/1176994266

Abstract

Let $\{X_i\}$ be a sequence of independent, identically distributed non-degenerate random variables taking values in $\mathbb{R}^d$ and $S_n = \sum^n_{i = 1} X_i, M_n = \max_{1\leqq i \leqq n} |S_i|$. Define for $x > 0, G(x) = P\{| X_1 | > x\}, K(x) = x^{-2}E(| X_1 |^2 1\{| X_1 | \leq x\}), M(x) = x^{-1} |E(X_1 1\{| X_1 | \leq x\})|,$ and $h(x) = G(x) + K(x) + M(x)$. Then if $\beta = \sup \{\alpha: \lim \sup x^\alpha h(x) = 0\}, \delta = \sup \{\alpha: \lim \inf x^\alpha h(x) = 0\}$, it is proved that $n^{-1/\alpha}M_n \rightarrow 0$ for $\alpha < \beta, \rightarrow \infty$ for $\alpha > \delta$, while the $\lim \inf$ is 0 and the $\lim \sup$ is $\infty$ for $\beta < \alpha < \delta$. Some alternative characterizations of the indices $\beta, \delta$ are obtained as well as the analogous results for Levy processes.

Citation

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William E. Pruitt. "The Growth of Random Walks and Levy Processes." Ann. Probab. 9 (6) 948 - 956, December, 1981. https://doi.org/10.1214/aop/1176994266

Information

Published: December, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0477.60033
MathSciNet: MR632968
Digital Object Identifier: 10.1214/aop/1176994266

Subjects:
Primary: 60F15

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 6 • December, 1981
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