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On the First Exit Time of a Nonnegative Markov Process Started at a Quasistationary Distribution

Part of:Sequential methodsMarkov processes

Published online by Cambridge University Press:  14 July 2016

Moshe Pollak and
Alexander G. Tartakovsky
Moshe Pollak*
Affiliation:
Hebrew University of Jerusalem
Alexander G. Tartakovsky*
Affiliation:
University of Southern California
*
∗ Postal address: Department of Statistics, Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email address: msmp@mscc.huji.ac.il
∗∗ Postal address: Department of Mathematics, University of Southern California, 3620 South Vermont Avenue, KAP-108, Los Angeles, CA 90089-2532, USA. Email address: tartakov@math.usc.edu
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Abstract

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Let {M n }n≥0 be a nonnegative time-homogeneous Markov process. The quasistationary distributions referred to in this note are of the form Q A (x) = limn→∞P(M n ≤ x | M 0 ≤ A,M 1 ≤ A, …, M n ≤ A). Suppose that M 0 has distribution Q A , and define T A Q A = min{n | M n > A, n ≥ 1}, the first time when M n exceeds A. We provide sufficient conditions for Q A (x) to be nonincreasing in A (for fixed x) and for T A Q A to be stochastically nondecreasing in A.

Keywords

Changepoint problemfirst exit timeMarkov processquasistationary distributionstationary distribution

MSC classification

Secondary:60J05: Discrete-time Markov processes on general state spaces60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)62L10: Sequential analysis62L15: Optimal stopping

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011, pp. 589 - 595
Copyright
Copyright © Applied Probability Trust 2011 

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