Asymptotic Exponentiality of the Distribution of First Exit Times for a Class of Markov Processes with Applications to Quickest Change Detection

Details Asymptotic Exponentiality of the Distribution of First Exit Times for a Class of Markov Processes with Applications to Quickest Change Detection Asymptotic Exponentiality of the Distribution of First Exit Times for a Class of Markov Processes with Applications to Quickest Change Detection

2006-09-28

arxiv.org/abs/math/0609780v2

Theory of Probability and Its Applications, vol. 53, issue 3, pp. 430-442, 2009

Mathematics

Probability

Scientific paper

We consider the first exit time of a nonnegative Harris-recurrent Markov process from the interval $[0,A]$ as $A\to\infty$. We provide an alternative method of proof of asymptotic exponentiality of the first exit time (suitably standardized) that does not rely on embedding in a regeneration process. We show that under certain conditions the moment generating function of a suitably standardized version of the first exit time converges to that of $\Exp(1)$, and we connect between the standardizing constant and the quasi-stationary distribution (assuming it exists). The results are applied to the evaluation of a distribution of run length to false alarm in change-point detection problems.

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