Summary
In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if S n is a random walk with negative mean and finite variance then there is a constant α so that (S [n.]/αn 1/2¦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 1=−a<0, ES 21 <∞, and there is a slowly varying function L so that P(S 1>x)∼x −q L(x) as x→∞ then (S [n.]/n¦S n >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.
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The research for this paper was started while the author was visiting W. Vervaat at the Katholieke Universiteit in Nijmegen, Holland, and was completed while the author was at UCLA being supported by funds from NSF grant MCS 77-02121
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Durrett, R. Conditioned limit theorems for random walks with negative drift. Z. Wahrscheinlichkeitstheorie verw Gebiete 52, 277–287 (1980). https://doi.org/10.1007/BF00538892
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DOI: https://doi.org/10.1007/BF00538892