This article is cited in 23 papers

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

M. Pollak^a, A. G. Tartakovskii<sup>b

a</sup> Hebrew University of Jerusalem
^b University of Southern California

Abstract: 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 Exponential (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.

Keywords: Markov process, stationary distribution, quasi-stationary distribution, first exit time, asymptotic exponentiality, change-point problems, CUSUM procedures, Shiryaev-Roberts procedures.

Received: 16.03.2007
Revised: 23.04.2008

Language: English

DOI: 10.4213/tvp2444

 English version: Theory of Probability and its Applications, 2009, 53:3, 430–442

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