Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-25T22:12:59.959Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant measures for Q-processes when Q is not regular

Published online by Cambridge University Press:  01 July 2016

P. K. Pollett
P. K. Pollett*
Affiliation:
University of Queensland
*
∗Postal address: Department of Mathematics, The University of Queensland, QLD 4072, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of determining invariant measures for continuous-time Markov chains directly from their transition rates is considered. The major result provides necessary and sufficient conditions for the existence of a unique ‘single-exit' chain with a specified invariant measure. This generalizes a result of Hou Chen-Ting and Chen Mufa that deals with symmetrically reversible chains. A simple sufficient condition for the existence of a unique honest chain for which the specified measure is invariant is also presented.

Keywords

MARKOV CHAINSSTATIONARY DISTRIBUTIONSBIRTH PROCESSES
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 2 , June 1991 , pp. 277 - 292
Copyright
Copyright © Applied Probability Trust 1991 

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

Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities , 2nd edn. Springer-Verlag, Berlin.Google Scholar
Hou, Chen-Ting and Chen, Mufa (1980) Markov processes and field theory. Kexue Tongbao 25, 807–811.Google Scholar
Kelly, F. P. (1983) Invariant measures and the q-matrix. In Probability, Statistics and Analysis , ed. Kingman, J. F. C. and Reuter, G. E. H., LMS Lecture Notes Series 79, Cambridge University Press, pp. 143–160.CrossRefGoogle Scholar
Kendall, D. G. (1956) Some further pathological examples in the theory of denumerable Markov processes. Quart. J. Math. Oxford 7, 39–56.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. 13, 417–431.CrossRefGoogle Scholar
Kendall, D. G. (1975) Some problems in mathematical genealogy. In Perspectives in Probability and Statistics, Papers in Honour of M. S. Bartlett , ed. Gani, J., Applied Probability Trust, Sheffield, 325–345.Google Scholar
Miller, R. G. Jr (1963) Stationary equations in continuous time Markov chains. Trans. Amer. Math. Soc. 109, 35–44.Google Scholar
Pollett, P. K. (1986) On the equivalence of µ-invariant measures for the minimal process and its q-matrix. Stoch. Proc. Appl. 22, 203–221.Google Scholar
Pollett, P. K. (1988) Reversibility, invariance and µ-invariance. Adv. Appl. Prob. 20, 600–621.Google Scholar
Pollett, P. K. (1990) A note on the classification of Q-processes when Q is not regular. J. Appl. Prob. 27, 278–290.Google Scholar
Pollett, P. K. (1991) On the construction problem for single-exit Markov chains. Bull. Austral. Math. Soc. Google Scholar
Reuter, G. E. H. (1957) Denumerable Markov processes and the associated contraction semigroups on l. Acta Math. 97, 1–46.CrossRefGoogle Scholar
Reuter, G. E. H. (1959) Denumerable Markov processes (II). J. London Math. Soc. 34, 81–91.Google Scholar
Reuter, G. E. H. (1962) Denumerable Markov processes (III). J. London Math. Soc. 37, 63–73.CrossRefGoogle Scholar
Reuter, G. E. H. (1967) Note on resolvents of denumerable submarkovian processes. Z. Wahrscheinlichkeitsth. 9, 16–19.Google Scholar
Tweedie, R. L. (1974) Some ergodic properties of the Feller minimal process. Quart. J. Math. Oxford 25, 485–495.Google Scholar
Vere-Jones, D. (1969) Some limit theorems for evanescent processes. Austral. J. Statist. 11, 67–78.Google Scholar
Williams, D. (1964) On the construction problem for Markov chains. Z. Wahrscheinlichkeitsth. 3, 227–246.Google Scholar
Xiangqum, Yang (1981) Construction of Q-processes satisfying Kolmogoroff's backward equations in single exit case or forward equations in single entrance case. Kexue Tongbao 26, 390–394.Google Scholar