On the Almost Sure Minimal Growth Rate of Partial Sum Maxima

Abstract

Let $S_n = X_1 + \cdots + X_n$ be partial sums of independent identically distributed random variables and let $a_n$ be an increasing sequence of positive constants tending to $\infty$. This paper concerns the almost sure lower limit of $\max_{1\leq j \leq n} S_j/a_n$. We prove that the lower limit is either 0 or $\infty$ under mild conditions and give integral tests to determine which is the case. Let $\tau = \inf\{n \geq 1: S_n > 0\}$ and $\tau_- = \inf\{n \geq 1: S_n \leq 0\}$. Several inequalities are given that determine up to scale constants various quantities involving truncated moments of the ladder variables $S_\tau$ and $\tau$ under three different conditions: $ES_\tau < \infty, E|S_{\tau-}| < \infty$ and $X$ symmetric. Moments of ladder variables are also discussed.