U. Einmahl et Dm. Mason, ON THE SMALLEST MAXIMAL INCREMENT OF PARTIAL-SUMS OF IID RANDOM-VARIABLES, Probability theory and related fields, 108(1), 1997, pp. 67-86
We study the almost sure Limiting behavior of the smallest maximal inc
rement of partial sums of n independent identically distributed random
variables for a variety of increment sizes k(n), where k(n) is a sequ
ence of integers satisfying 1 less than or equal to k(n) less than or
equal to n, and going to infinity at various rates. Our aim is to obta
in universal results on such behavior under little or no assumptions o
n the underlying distribution function.