Geodesic
Asymptotic Exponentiality of the Distribution of First Exit Times for a Class of Markov Processes with Applications to Quickest Change Detection
Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 3, pp. 500-515
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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. 
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M. Pollak; A. G. Tartakovskii. Asymptotic Exponentiality of the Distribution of First Exit Times for a Class of Markov Processes with Applications to Quickest Change Detection. Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 3, pp. 500-515. http://geodesic.mathdoc.fr/item/TVP_2008_53_3_a4/

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