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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.
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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 

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