Level-crossings of symmetric random walks and their application

Abstract

Let X1, X2, ... be a sequence of independently and identically distributed random variables with EX1=0, and let S0=0 and St=St-1+Xt, t=1,2,..., be a random walk. Denote τ=cases∈f\t>1: St≤0\, &if \ X1>0, 1, &otherwise. cases Let α denote a positive number, and let Lα denote the number of level-crossings from the below (or above) across the level α during the interval [0, τ]. Under quite general assumption, an inequality for the expected number of level-crossings is established. Under some special assumptions, it is proved that there exists an infinitely increasing sequence αn such that the equality ELαn=cP\X1>0\ is satisfied, where c is a specified constant that does not depend on n. The result is illustrated for a number of special random walks. We also give non-trivial examples from queuing theory where the results of this theory are applied.