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

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 $S_n$.