Conditioned limit theorems for random walks with negative drift

In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if Sn is a random walk with negative mean and finite variance then there is a constant α so that (S[n.]/αn1/2|N>n) converges weakly to a process which he called the Brownian excursion. It will be shown that his conjecture is false or, more precisely, that if ES1=-a<0, ES12<∞, and there is a slowly varying function L so that P(S1>x)∼x-q L(x) as x→∞ then (S[n.]/n|Sn>0) and (S[n.]/n|N>n) converge weakly to nondegenerate limits. The limit processes have sample paths which have a single jump (with d.f. (1-(x/a)-q)+) and are otherwise linear with slope -a. The jump occurs at a uniformly distributed time in the first case and at t=0 in the second. © 1980 Springer-Verlag.

Duke Scholars

Author Richard Timothy Durrett Mathematics