The Annals of Probability

Limit Theorems for Random Walks Conditioned to Stay Positive

Robert W. Keener

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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

Permanent link to this document
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
links.jstor.org

Subjects
Primary: 60J15
Secondary: 60G50: Sums of independent random variables; random walks

Keywords
Large deviations Markov chains conditional limit theorems quasistationary distributions

Citation

Keener, Robert W. Limit Theorems for Random Walks Conditioned to Stay Positive. Ann. Probab. 20 (1992), no. 2, 801--824. doi:10.1214/aop/1176989807. https://projecteuclid.org/euclid.aop/1176989807


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