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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.

References

[1]Bahadur, R. R. and Rao, R. R. (1960) On deviations of the sample mean. Ann. Math. Statist. 31, 1015–1027.Google Scholar
[2]Baxter, G. (1961) An analytic approach to finite fluctuation problems in probability. J. Analyse Math. 9, 31–70.Google Scholar
[3]Chung, K. L. (1968) A Course in Probability Theory. Harcourt, Brace and World, New York.Google Scholar
[4]Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
[5]Daley, D. (1968) Quasi-stationary behavior of a left-continuous random walk. Ann. Math. Statist. 40, 532–539.Google Scholar
[6]Feller, W. (1971) An Introduction to Probability Theory and its Applications. Vol. 2. John Wiley, New York.Google Scholar
[7]Heathcote, C. R. (1967) Complete exponential convergence and related topics. J. Appl. Prob. 4, 217–256.Google Scholar
[8]Iglehart, D. L. (1974) Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab. To appear.Google Scholar
[9]Kennedy, D. P. (1974) Limiting diffusions for the conditioned M/G/l queue. J. Appl. Prob. 11, 355–362.Google Scholar
[10]Kyprianou, E. K. (1971) On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals. J. Appl. Prob. 8, 494–507.Google Scholar
[11]Kyprianou, E. K. (1972a) On the quasi-stationary distributions of the GI/M/1 queue. J. Appl. Prob. 9, 117–128.Google Scholar
[12]Kyprianou, E. K. (1972b) The quasi-stationary distributions of queues in heavy traffic. J. Appl. Prob. 9, 821–831.Google Scholar
[13]Pollaczek, F. (1952) Fonctions caractéristiques de certaines répartitions définies au moyen de la notion d'ordre. Application à la théorie des attentes. C. R. Acad. Sci. Paris. A-B 234, 2334–2336.Google Scholar
[14]Prabhu, N. U. (1965) Queues and Inventories. John Wiley, New York.Google Scholar
[15]Seneta, E. and Vere-Jones, D. (1967) On quasi-stationary distributions in discrete time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434.Google Scholar