Math @ Duke

Publications [#243432] of Richard T. Durrett
Papers Published
 Durrett, R, Conditioned limit theorems for random walks with negative drift,
Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete, vol. 52 no. 3
(January, 1980),
pp. 277287, Springer Nature, ISSN 00443719 [doi]
(last updated on 2019/06/16)
Abstract: 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.]/αn1/2N>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)∼xq L(x) as x→∞ then (S[n.]/nSn>0) and (S[n.]/nN>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 SpringerVerlag.


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