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Spectral structure of the first-passage-time densities for classes of Markov chains

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

Keilson [7] showed that for a birth-death process defined on non-negative integers with reflecting barrier at 0 the first-passage-time density from 0 to N (N to N + 1) has Pólya frequency of order infinity (is completely monotone). Brown and Chaganty [3] and Assaf et al. [1] studied the first-passage-time distribution for classes of discrete-time Markov chains and then produced the essentially same results as these through a uniformization. This paper addresses itself to an extension of Keilson's results to classes of Markov chains such as time-reversible Markov chains, skip-free Markov chains and birth-death processes with absorbing states. The extensions are due to the spectral representations of the infinitesimal generators governing these Markov chains. Explicit densities for those first-passage times are also given.

Keywords

PÓLYA FREQUENCY OF ORDER INFINITYCOMPLETELY MONOTONESPECTRAL REPRESENTATIONSSKIP-FREE MARKOV CHAINTIME-REVERSIBLE MARKOV CHAINBIRTH-DEATH PROCESS WITH ABSORBING STATES
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 3 , September 1987 , pp. 631 - 643
Copyright
Copyright © Applied Probability Trust 1987 

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References

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