The Annals of Probability

On Conditioning a Random Walk to Stay Nonnegative

J. Bertoin and R. A. Doney

Full-text: Open access

Abstract

Let $S$ be a real-valued random walk that does not drift to $\infty$, so $P(S_k \geq 0$ for all $k) = 0$. We condition $S$ to exceed $n$ before hitting the negative half-line, respectively, to stay nonnegative up to time $n$. We study, under various hypotheses, the convergence of these conditional laws as $n \rightarrow \infty$. First, when $S$ oscillates, the two approximations converge to the same probability law. This feature may be lost when $S$ drifts to $-\infty$. Specifically, under suitable assumptions on the upper tail of the step distribution, the two approximations then converge to different probability laws.

Article information

Source
Ann. Probab., Volume 22, Number 4 (1994), 2152-2167.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988497

Digital Object Identifier
doi:10.1214/aop/1176988497

Mathematical Reviews number (MathSciNet)
MR1331218

Zentralblatt MATH identifier
0834.60079

JSTOR
links.jstor.org

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

Keywords
Random walk conditional law $h$-transform ladder variable limit theorems

Citation

Bertoin, J.; Doney, R. A. On Conditioning a Random Walk to Stay Nonnegative. Ann. Probab. 22 (1994), no. 4, 2152--2167. doi:10.1214/aop/1176988497. https://projecteuclid.org/euclid.aop/1176988497


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