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October 1996 Hitting probabilities and large deviations
Jeffrey F. Collamore
Ann. Probab. 24(4): 2065-2078 (October 1996). DOI: 10.1214/aop/1041903218

Abstract

Let ${Y_n}_{n \in \mathbb{Z}_+}$ be a sequence of random variables in $\mathbb{R}^d$ and let $A \subset \mathbb{R}^d$. Then $\mathbf{P}\{Y_n \in A \text{for some $n$}\}$ is the hitting probability of the set A by the sequence ${Y_n}$. We consider the asymptotic behavior, as $m \to \infty$, of $\mathbf{P}\{Y_n \in mA \text{some $n$}\} = \mathbf{P}{\text{hitting $mA$}$ whenever (1) the probability law of $Y_n/n$ satisfies the large deviation principle and (2) the central tendency of $Y_n/n$ is directed away from the given set A. For a particular function $\tilde{I}$, we show $m \to \infty$, of $\mathbf{P}\{Y_n \in mA \text{some $n$}\} \approx \exp (-m \tilde{I}(A))$.

Citation

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Jeffrey F. Collamore. "Hitting probabilities and large deviations." Ann. Probab. 24 (4) 2065 - 2078, October 1996. https://doi.org/10.1214/aop/1041903218

Information

Published: October 1996
First available in Project Euclid: 6 January 2003

zbMATH: 0879.60021
MathSciNet: MR1415241
Digital Object Identifier: 10.1214/aop/1041903218

Subjects:
Primary: 60F10
Secondary: 60K10

Keywords: hitting probabilities , large deviations

Rights: Copyright © 1996 Institute of Mathematical Statistics

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Vol.24 • No. 4 • October 1996
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