## The Annals of Probability

### A universal form of the Chung-type law of the iterated logarithm

Harry Kesten

#### 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 $S_n$.

#### Article information

Source
Ann. Probab., Volume 25, Number 4 (1997), 1588-1620.

Dates
First available in Project Euclid: 7 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1023481104

Digital Object Identifier
doi:10.1214/aop/1023481104

Mathematical Reviews number (MathSciNet)
MR1487429

Zentralblatt MATH identifier
0903.60053

Subjects
Primary: 60J15 60F15

#### Citation

Kesten, Harry. A universal form of the Chung-type law of the iterated logarithm. Ann. Probab. 25 (1997), no. 4, 1588--1620. doi:10.1214/aop/1023481104. https://projecteuclid.org/euclid.aop/1023481104

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