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Random walks with negative drift conditioned to stay positive

Published online by Cambridge University Press:  14 July 2016

Donald L. Iglehart
Donald L. Iglehart*
Affiliation:
Stanford University
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Abstract

Let {Xk: k ≧ 1} be a sequence of independent, identically distributed random variables with EX1 = μ < 0. Form the random walk {Sn: n ≧ 0} by setting S0 = 0, Sn = X1 + … + Xn, n ≧ 1. Let T denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of X1) that Sn, conditioned on T > n converges weakly to a limit random variable, S∗, and to find the Laplace transform of the distribution of S∗. We also investigate a collection of random walks with mean μ < 0 and conditional limits S∗ (μ), and show that S∗ (μ), properly normalized, converges to a gamma distribution of second order as μ ↗ 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.

Keywords

CONDITIONAL LIMIT THEOREMSGI/G/1 QUEUEQUASI-STATIONARY DISTRIBUTIONSRANDOM WALKS
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 4 , December 1974 , pp. 742 - 751
Copyright
Copyright © Applied Probability Trust 1974 

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Footnotes

Research supported by N.S.F. Grant GP-31392X1 and Office of Naval Research Contract N00014-67-A-0112-0031.

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