Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-07-04T05:54:50.162Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  14 July 2016

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

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.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1987 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Assaf, D., Shaked, M. and Shanthikumar, J. G. (1985) First-passage-times with PFr densities J. Appl. Prob. 22, 185196.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, 10001008.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, 283285.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, 391398.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