Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T05:15:13.219Z Has data issue: false hasContentIssue false
  • English
  • Français

Similar Markov chains

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

P. K. Pollett
P. K. Pollett*
Affiliation:
University of Queensland
*
1Postal address: Dept Mathematics, The University of Queensland Qld 4072, Australia. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Lenin et al. (2000) recently introduced the idea of similarity in the context of birth-death processes. This paper examines the extent to which their results can be extended to arbitrary Markov chains. It is proved that, under a variety of conditions, similar chains are strongly similar in a sense which is described, and it is shown that minimal chains are strongly similar if and only if the corresponding transition-rate matrices are strongly similar. A general framework is given for constructing families of strongly similar chains; it permits the construction of all such chains in the irreducible case.

Keywords

Invariant measuresbirth-death processessimilar Markov chains

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J35: Transition functions, generators and resolvents
Type
Markov chains
Information
Journal of Applied Probability , Volume 38 , Issue A: Probability, Statistics and Seismology , 2001 , pp. 53 - 65
Copyright
Copyright © Applied Probability Trust 2001 

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

Anderson, W. J. (1991). Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer, New York.CrossRefGoogle Scholar
Darroch, J. N. and Seneta, E. (1965). On quasi-stationary distributions in absorbing discrete-time Markov chains. J. Appl. Probab. 2, 88–100.CrossRefGoogle Scholar
Darroch, J. N. and Seneta, E. (1967). On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Probab. 4, 192–196.CrossRefGoogle Scholar
Di Crescenzo, A. (1994a). On certain transformation properties of birth-and-death processes. In Cybernetics and Systems '94 , ed. Trappl, R., World-Scientific, Singapore, 839–846.Google Scholar
Di Crescenzo, A. (1994b). On some transformations of bilateral birth-and-death processes with applications to first passage time evaluations. In Proc. 17th Symp. Inf. Theory Applic. (SITA '94) , December 6-9, Hiroshima, Japan, 739–742.Google Scholar
Elmes, S., Pollett, P. and Walker, D. (2000). Further results on the relationship between µ-invariant measures and quasi-stationary distributions for absorbing continuous-time Markov chains. Math. Comput. Modell. 31, 107–113.CrossRefGoogle Scholar
Kelly, F. P. (1979). Reversibility and Stochastic Networks. Wiley, Chichester.Google Scholar
Kelly, F. P. (1983). Invariant measures and the q-matrix. In Probability, Statistics and Analysis (London Math. Soc. Lecture Notes 79), eds Kingman, J. F. C. and Reuter, G. E. H., Cambridge University Press, 143–160.CrossRefGoogle Scholar
Kingman, J. F. C. (1963). The exponential decay of Markov transition probabilities. Proc. London Math. Soc. 13, 337–358.CrossRefGoogle Scholar
Lenin, R. B., Parthasarathy, P. R., Scheinhardt, W. R. W. and Van Doorn, E. A. (2000). Families of birth-death processes with similar time-dependent behaviour. J. Appl. Probab. 37, 835–849.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.CrossRefGoogle Scholar
Pollett, P. K. (2000). Quasistationary Distributions: A Bibliography. Statistics Research Report, Department of Mathematics, The University of Queensland. Available at http://www.maths.uq.edu.au/~pkp/papers/qsds.html.Google Scholar
Robert, C. P. and Casella, G. (1999). Monte Carlo Statistical Methods. Springer, New York.CrossRefGoogle Scholar
Seneta, E. (1966). Quasi-stationary behaviour in the random walk with continuous time. Austral. J. Statist. 8, 92–98.CrossRefGoogle Scholar
Seneta, E. and Vere-Jones, D. (1966). On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probab. 3, 403–434.CrossRefGoogle Scholar
Tuominen, P. and Tweedie, R. L. (1994). Subgeometric rates of convergence of f-ergodic Markov chains. Adv. Appl. Probab. 26, 775–798.CrossRefGoogle Scholar
Vere-Jones, D. (1962). Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford (2) 13, 7–28.CrossRefGoogle Scholar
Vere-Jones, D. (1963). On the spectra of some linear operators associated with queueing systems. Z. Wahrscheinlichkeitsth. 2, 12–21.CrossRefGoogle Scholar
Vere-Jones, D. (1967). Ergodic properties of non-negative matrices, I. Pacific J. Math. 22, 361–386.CrossRefGoogle Scholar
Vere-Jones, D. (1968). Ergodic properties of non-negative matrices, II. Pacific J. Math. 26, 601–620.CrossRefGoogle Scholar
Walker, D. M. (1998). The expected time until absorption when absorption is not certain. J. Appl. Probab. 35, 812–823.CrossRefGoogle Scholar
Wang, Zikun and Yang, Xiangqun (1992). Birth and Death Processes and Markov Chains. Springer, Berlin, and Science Press, Beijing.Google Scholar