Moments of ladder heights in random walks

Abstract

A well-known result in the theory of random walks states that E{X 2} is finite if and only if E{Z+ } and E{Z_} are both finite (Z + and Z_ being the ladder heights and X a typical step-length) in which case E{X 2} = 2E{Z+ }E{Z_}. This paper contains results relating the existence of moments of X of order ß to the existence of the moments of Z + and Z_ of order ß – 1. The main result is that if β > 2 E{|X|β} < ∞ if and only if and are both finite.