A Universal Chung-Type Law of the Iterated Logarithm

Abstract

Let $X_1, X_2,\ldots$, be a sequence of independent and identically distributed random variables. We find sequences of norming and centering constants $\alpha_n$ and $\beta_n$ such that a universal Chung-type law of the iterated logarithm holds, namely, $\lim \inf_{n\rightarrow \infty} \max_{1\leq k \leq n}|S_k - k\beta_n|/\alpha_n < \infty$ almost surely, where $S_k$ denotes the sum of the first $k$ of $X_1, X_2,\ldots, k \geq 1$. If the underlying distribution function is in the Feller class, we show that this $\lim \inf$ is strictly positive with probability 1.