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On passage and conditional passage times for Markov chains in continuous time

Published online by Cambridge University Press:  14 July 2016

Masaaki Kijima
Masaaki Kijima*
Affiliation:
Tokyo Institute of Technology
*
∗Postal address: Department of Information Sciences, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, Japan.
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Abstract

Let X(t) be a temporally homogeneous irreducible Markov chain in continuous time defined on . For k < i < j, let H = {k + 1, ···, j − 1} and let kTij (jTik) be the upward (downward) conditional first-passage time of X(t) from i to j(k) given no visit to . These conditional passage times are studied through first-passage times of a modified chain HX(t) constructed by making the set of states absorbing. It will be shown that the densities of kTij and jTik for any birth-death process are unimodal and the modes kmij (jmik) of the unimodal densities are non-increasing (non-decreasing) with respect to i. Some distribution properties of kTij and jTik for a time-reversible Markov chain are presented. Symmetry among kTij, jTik, and is also discussed, where , and are conditional passage times of the reversed process of X(t).

Keywords

FIRST PASSAGE TIMECONDITIONAL FIRST PASSAGE TIMEUNIMODALITYSYMMETRYREVERSED MARKOV CHAIN
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 2 , June 1988 , pp. 279 - 290
Copyright
Copyright © Applied Probability Trust 1988 

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