Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-29T08:34:48.965Z Has data issue: false hasContentIssue false
  • English
  • Français

On the representation of symmetric transition functions

Published online by Cambridge University Press:  01 July 2016

W. J. Anderson  and
P. M. McDunnough
W. J. Anderson*
Affiliation:
McGill University
P. M. McDunnough*
Affiliation:
University of Toronto
*
∗Postal address: Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke St West, Montreal, Canada H3A 2K6.
∗∗Postal address: Department of Statistics, University of Toronto, Toronto, Ontario, Canada M5S 1A1.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we give an alternative derivation of Kendall's representation for symmetric transition functions which relies on the backward and/or forward integral recursions. The proof uses a lemma concerning approximation by finite sections (which is useful in its own right) and is similar to the original proof for birth and death processes by Lederman and Reuter. Finally, we obtain a general result guaranteeing the existence of representations of transition functions such as those obtained by Pruitt and Iglehart.

Keywords

CONTINUOUS-TIME MARKOV CHAINSREVERSIBLEORTHOGONALq-MATRIX
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 3 , September 1990 , pp. 548 - 563
Copyright
Copyright © Applied Probability Trust 1990 

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

Billingsley, P. (1979) Probability and Measure. Wiley, New York.Google Scholar
Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, Berlin.Google Scholar
Iglehart, D. L. (1964) Reversible competition processes. Z. Wahrscheinlichkeitsth. 2, 314–331.CrossRefGoogle Scholar
Karlin, S. and Mcgregor, J. L. (1957a) The differential equations of birth and death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–546.CrossRefGoogle Scholar
Karlin, S. and Mcgregor, J. (1957b) The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.CrossRefGoogle Scholar
Karlin, S. and Mcgregor, J. (1958a) Linear growth birth and death processes. J. Math. Mech. 7, 643–662.Google Scholar
Karlin, S. and Mcgregor, J. (1958b) Many server queueing processes with Poisson input and exponential service times. Pacific J. Math. 8, 87–118.Google Scholar
Karlin, S. and Mcgregor, J. (1959a) Coincidence properties of birth and death processes. Pacific J. Math. 9, 1109–40.Google Scholar
Karlin, S. and Mcgregor, J. (1959b) Coincidence probabilities. Pacific J. Math. 9, 1141–64.Google Scholar
Karlin, S. and Mcgregor, J. (1965) Ehrenfest urn models. J. Appl. Prob. 2, 352–376.CrossRefGoogle Scholar
Karlin, S. and Tavare, S. (1982) Linear birth and death processes with killing. J. Appl. Prob. 19, 477–487.Google Scholar
Keilson, J. (1979) Markov Chain Models—Rarity and Exponentiality. Springer-Verlag, Berlin.Google Scholar
Kendall, D. G. (1959) Unitary dilations of one-parameter semigroups of Markov transition operators, and the corresponding integral representations for Markov processes with a countable infinity of states. Proc. London. Math. Soc. (3) 9, 417–431.Google Scholar
Ledermann, W. and Reuter, G. E. H. (1954) Spectral theory for the differential equations of simple birth and death processes. Phil. Trans. Roy. Soc. London A 246, 321–369.Google Scholar
Pruitt, W. E. (1963) Bilateral birth and death processes. Trans. Amer. Math. Soc. 107, 508–525.CrossRefGoogle Scholar
Reuter, G. E. H. (1957) Denumerable Markov processes and the associated contraction semigroups on l . Acta Math. 97, 1–46.Google Scholar
Reuter, G. E. H. (1961) Competition processes. Proc. 4th Berkeley Symp. Math. Statist. Prob. 2, 421–430.Google Scholar
Van Doorn, E. A. (1980) Stochastic Monotonicity and Queueing Applications of Birth-Death Processes. Lecture Notes in Statistics 4, Springer-Verlag, New York.Google Scholar