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October 1997 A universal form of the Chung-type law of the iterated logarithm
Harry Kesten
Ann. Probab. 25(4): 1588-1620 (October 1997). DOI: 10.1214/aop/1023481104

Abstract

Let $\{X_i\}_ {i \geq 1}$ be i.i.d. random variables with common distribution function $F$, and let $S_n=\sum_1^n X_i$. We find a necessary and sufficient condition (directly in terms of$F$) for the existence of sequences of constants $\{\alpha_n\}$ and $\{\beta_n\}$ with $\beta_n\uparrow\infty$ such that $0<\liminf \beta_n^{-1}\max_{j\leq n}|S_j- \alpha_j|<\infty$ w.p.1. and such that for any choice of $\tilde{\alpha}_n$, it holds w.p.1 that $\liminf \beta_n^{-1}\max_{j\leqn|S_j - \tilde{\alpha}_j|>0$. The latter requirement is added to rule out sequences ${\beta_n}$ which grow too fast and entirely overwhelm the fluctuations of $S_n$.

Citation

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Harry Kesten. "A universal form of the Chung-type law of the iterated logarithm." Ann. Probab. 25 (4) 1588 - 1620, October 1997. https://doi.org/10.1214/aop/1023481104

Information

Published: October 1997
First available in Project Euclid: 7 June 2002

zbMATH: 0903.60053
MathSciNet: MR1487429
Digital Object Identifier: 10.1214/aop/1023481104

Subjects:
Primary: 60F15 , 60J15

Keywords: Chung-type law of the iterated logarithm , Law of the iterated logarithm , sums of i.i.d. random variables

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.25 • No. 4 • October 1997
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