Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-18T01:56:28.860Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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