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 c~ so that (St,.j/c~nl/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 ES~ = a < 0 , ES~ < oo, and there is a slowly varying function L so that P(SI > x)~ x -q L(x) as x -~ oo then (SE,,.j/nlS,,>O) and (St,,jnlN>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.