The Annals of Probability

Limit Theorems for Random Walks Conditioned to Stay Positive

Robert W. Keener

Abstract

Let $\{S_n\}$ be a random walk on the integers with negative drift, and let $A_n = \{S_k \geq 0, 1 \leq k \leq n\}$ and $A = A_\infty$. Conditioning on $A$ is troublesome because $P(A) = 0$ and there is no natural sigma-field of events "like" $A. A$ natural definition of $P(B\mid A)$ is $\lim_{n\rightarrow\infty}P(B\mid A_n)$. The main result here shows that this definition makes sense, at least for a large class of events $B$: The finite-dimensional conditional distributions for the process $\{S_k\}_{k\geq 0}$ given $A_n$ converge strongly to the finite-dimensional distributions for a measure $\mathbf{Q}$. This distribution $\mathbf{Q}$ is identified as the distribution for a stationary Markov chain on $\{0,1,\ldots\}$.

Article information

Source
Ann. Probab., Volume 20, Number 2 (1992), 801-824.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176989807

Digital Object Identifier
doi:10.1214/aop/1176989807

Mathematical Reviews number (MathSciNet)
MR1159575

Zentralblatt MATH identifier
0756.60062

JSTOR