Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-15T11:19:07.812Z Has data issue: false hasContentIssue false
  • English
  • Français

Limiting diffusion for random walks with drift conditioned to stay positive

Published online by Cambridge University Press:  14 July 2016

Peichuen Kao
Peichuen Kao*
Affiliation:
Systems Management Center, University of Southern California
*
∗Now at Temple University, Philadelphia, Pa.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let {ξk : k ≧ 1} be a sequence of independent, identically distributed random variables with E{ξ1} = μ ≠ 0. Form the random walk {Sn : n ≧ 0} by setting S0, Sn = ξ1 + ξ2 + ··· + ξn, n ≧ 1. Define the random function Xn by setting where α is a norming constant. Let N 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 ξ1) that the finite-dimensional distributions of Xn, conditioned on n < N < ∞ converge to those of the Brownian excursion process.

Keywords

CONDITIONAL LIMIT THEOREMSRANDOM WALKSBROWNIAN EXCURSION PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 2 , June 1978 , pp. 280 - 291
Copyright
Copyright © Applied Probability Trust 1978 

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.)

References

Bahadur, R. R. and Rao, R. R. (1960) On deviations of the sample mean. Ann. Math. Statist. 31, 1015–1027.Google Scholar
Belkin, B. (1972) An invariance principle for conditioned random walk attracted to a stable law. Z. Wahrscheinlichkeitsth. 21, 45–64.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Borovkov, A. A. (1965) New limit theorems in boundary problems for sums of independent terms. Selected Trans. Math. Statist. Prob. 5, 315–372.Google Scholar
Chung, K. L. (1968) A Course of Probability Theory. Harcourt, Brace and World, New York.Google Scholar
Iglehart, D. L. (1974a) Functional central limit theorems for random walks conditioned to stay positive. Ann Prob. 2, 608–619.Google Scholar
Iglehart, D. L. (1974b) Random walks with negative drift conditioned to stay positive. J. Appl. Prob. 11, 742–751.CrossRefGoogle Scholar
Ito, K. and Mckean, J. P. (1965) Diffusion Processes and their Sample Paths. Springer-Verlag, Berlin.Google Scholar
Kao, P. (1976) Conditioned Limit Theorems in Queueing Theory. Ph.D. Thesis, Department of Operations Research, Stanford University.Google Scholar
Kennedy, D. P. (1974) Limiting diffusions for the conditioned M/G/1 queue. J. Appl. Prob. 11, 355–362.Google Scholar
Veraverbeke, N. (1977) Asymptotic behavior of Wiener–Hopf factors of a random walk. Stoch. Proc. Appl. 5, 27–37.Google Scholar
Veraverbeke, N. and Teugels, J. L. (1975) The exponential rate of convergence of the distribution of the maximum of a random walk. J. Appl. Prob. 12, 279–288.Google Scholar