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April, 1992 Limit Theorems for Random Walks Conditioned to Stay Positive
Robert W. Keener
Ann. Probab. 20(2): 801-824 (April, 1992). DOI: 10.1214/aop/1176989807


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 $\{0,1,\ldots\}$.


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Robert W. Keener. "Limit Theorems for Random Walks Conditioned to Stay Positive." Ann. Probab. 20 (2) 801 - 824, April, 1992.


Published: April, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0756.60062
MathSciNet: MR1159575
Digital Object Identifier: 10.1214/aop/1176989807

Primary: 60J15
Secondary: 60G50

Keywords: conditional limit theorems , large deviations , Markov chains , quasistationary distributions

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 2 • April, 1992
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