Conditioned limit theorems for random walks with negative drift

  • Richard Durrett


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.]n1/2¦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, ES 1 2 <∞, and there is a slowly varying function L so that P(S1>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.


Stochastic Process Random Walk Probability Theory Limit Theorem Mathematical Biology 
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Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Richard Durrett
    • 1
  1. 1.University of CaliforniaLos AngelesUSA

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