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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

[1] Assaf, D., Shaked, M. and Shanthikumar, J. G. (1985) First-passage-times with PFr densities J. Appl. Prob. 22, 185–196.Google Scholar
[2] Bellman, R. E. (1960) Introduction to Matrix Analysis. McGraw-Hill, New York.Google Scholar
[3] Brown, M. and Chaganty, N. R. (1983) On the first passage time distribution for a class of Markov chains. Ann. Prob. 11, 1000–1008.Google Scholar
[4] Derman, C., Ross, S. M. and Schechner, Z. (1983) A note on first passage times in birth and death and nonnegative diffusion processes. Naval Res. Logist. Quart. 30, 283–285.Google Scholar
[5] Karlin, S. (1968) Total Positivity. Stanford University Press, Stanford, Ca.Google Scholar
[6] Keilson, J. (1965) Green's Function Methods in Probability Theory. Hafner, New York.Google Scholar
[7] Keilson, J. (1971) Log-concavity and log-convexity in passage time densities of diffusion and birth-death processes. J. Appl. Prob. 8, 391–398.Google Scholar
[8] Keilson, J. (1979) Markov Chain Models — Rarity and Exponentiality. Springer-Verlag, New York.CrossRefGoogle Scholar
[9] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
[10] Kemeny, J. G. and Snell, J. L. (1960) Finite Markov Chains. Von Nostrand Reinhold Company, New York.Google Scholar
[11] Senata, E. (1981) Non-negative Matrices and Markov Chains, 2nd edn. Springer-Verlag, New York.Google Scholar