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Conditioned limit theorems for random walks with negative drift

  • Richard Durrett
Article

Summary

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.

Keywords

Stochastic Process Random Walk Probability Theory Limit Theorem Mathematical Biology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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