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.
@article{TVP_2008_53_3_a4,
author = {M. Pollak and A. G. Tartakovskii},
title = {Asymptotic {Exponentiality} of the {Distribution} of {First} {Exit} {Times} for a {Class} of {Markov} {Processes} with {Applications} to {Quickest} {Change} {Detection}},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {500--515},
publisher = {mathdoc},
volume = {53},
number = {3},
year = {2008},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TVP_2008_53_3_a4/}
}
TY - JOUR AU - M. Pollak AU - A. G. Tartakovskii TI - Asymptotic Exponentiality of the Distribution of First Exit Times for a Class of Markov Processes with Applications to Quickest Change Detection JO - Teoriâ veroâtnostej i ee primeneniâ PY - 2008 SP - 500 EP - 515 VL - 53 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TVP_2008_53_3_a4/ LA - en ID - TVP_2008_53_3_a4 ER -
%0 Journal Article %A M. Pollak %A A. G. Tartakovskii %T Asymptotic Exponentiality of the Distribution of First Exit Times for a Class of Markov Processes with Applications to Quickest Change Detection %J Teoriâ veroâtnostej i ee primeneniâ %D 2008 %P 500-515 %V 53 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/item/TVP_2008_53_3_a4/ %G en %F TVP_2008_53_3_a4
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/