On the smallest maximal increment of partial sums of i.i.d. random variables

Abstract

We study the almost sure limiting behavior of the smallest maximal increment of partial sums of \(n\) independent identically distributed random variables for a variety of increment sizes \(k_n\), where \(k_n\) is a sequence of integers satisfying \(1 \le k_n \le n\), and going to infinity at various rates. Our aim is to obtain universal results on such behavior under little or no assumptions on the underlying distribution function.