Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-26T10:23:38.102Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes Sequential methods

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let {Mn}n≥0 be a nonnegative time-homogeneous Markov process. The quasistationary distributions referred to in this note are of the form QA(x) = limn→∞P(Mn ≤ x | M0 ≤ A, M1 ≤ A, …, Mn ≤ A). Suppose that M0 has distribution QA, and define TAQA = min{n | Mn > A, n ≥ 1}, the first time when Mn exceeds A. We provide sufficient conditions for QA(x) to be nonincreasing in A (for fixed x) and for TAQA 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 spaces 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 62L10: Sequential analysis 62L15: Optimal stopping
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 589 - 595
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Borovkov, A. A. (1976). Stochastic Processes in Queuing Theory. Springer, New York.CrossRefGoogle Scholar
[2] Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
[3] Moustakides, G. V., Polunchenko, A. S. and Tartakovsky, A. G. (2011). A numerical approach to performance analysis of quickest change-point detection procedures. Statistica Sinica 21, 571–598.Google Scholar
[4] Pollak, M. (1985). Optimal detection of a change in distribution. Ann. Statist. 13, 206–227.Google Scholar
[5] Pollak, M. and Siegmund, D. (1986). Convergence of quasistationary to stationary distributions for stochastically monotone Markov processes. J. Appl. Prob. 23, 215–220.Google Scholar
[6] Pollett, P. K. (2008). Quasi-stationary distributions: a bibliography. Available at www.maths.uq.edu.au/∼pkp/papers/qsds/qsds.pdf.Google Scholar
[7] Tartakovsky, A. G., Pollak, M. and Polunchenko, A. S. (2011). Third-order asymptotic optimality of the generalized Shiryaev–Roberts changepoint detection procedures. To appear in Theory Prob. Appl. Google Scholar