Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-29T10:51:11.624Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasistationarity of continuous-time Markov chains with positive drift

Published online by Cambridge University Press:  17 February 2009

Pauline Coolen-Schrijner ,
Andrew Hart  and
Phil Pollett
Pauline Coolen-Schrijner
Affiliation:
Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, UK.
Andrew Hart
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld 4001, Australia.
Phil Pollett
Affiliation:
Department of Mathematics, The University of Queensland, Qld 4072, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We shall study continuous-time Markov chains on the nonnegative integers which are both irreducible and transient, and which exhibit discernible stationarity before drift to infinity “sets in”. We will show how this ‘quasi’ stationary behaviour can be modelled using a limiting conditional distribution: specifically, the limiting state probabilities conditional on not having left 0 for the last time. By way of a dual chain, obtained by killing the original process on last exit from 0, we invoke the theory of quasistationarity for absorbing Markov chains. We prove that the conditioned state probabilities of the original chain are equal to the state probabilities of its dual conditioned on non-absorption, thus allowing to establish the simultaneous existence and then equivalence, of their limiting conditional distributions. Although a limiting conditional distribution for the dual chain is always quasistationary distribution in the usual sense, a similar statement is not possible for the original chain.

Type
Research Article
Information
The ANZIAM Journal , Volume 41 , Issue 4 , April 2000 , pp. 423 - 441
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Anderson, W. J., Continuous-Time Markov Chains: An Applications-Oriented Approach (Springer, New York, 1991).CrossRefGoogle Scholar
[2]Breyer, L. A. and Hart, A. G., “Approximations of quasistationary distributions for Markov chains”, Math. Computer Modelling 25 (1999) to appear.Google Scholar
[3]Coolen-Schrijner, P. and Pollett, P. K., “Quasi-stationarity of discrete-time Markov chains with drift to infinity”, Methodol. Comput. Appl. Probab. 1 (1999) 81–96.CrossRefGoogle Scholar
[4]Darroch, J. N. and Seneta, E., “On quasi-stationary distributions in absorbing continuous-time finite Markov chains”, J. Appl. Probab. 4 (1967) 192–196.CrossRefGoogle Scholar
[5]Golub, H. G. and van Loan, C. F., Matrix Computations, 3rd ed. (Johns Hopkins University Press, Baltimore, 1993).Google Scholar
[6]Hart, A. G., “Quasistationary distributions for continuous-time Markov chains”, Ph. D. Thesis, Department of Mathematics, The University of Queensland, 1997.Google Scholar
[7]Hart, A. G. and Tweedie, R. L., “Convergence of invariant measures of truncation approximations to Markov processes” (1998) submitted for publication.Google Scholar
[8]Kesten, H., “A ratio limit theorem for (sub) Markov chains on {1, 2, 3 …} with bounded jumps”, Adv. Appl. Probab. 27 (1995) 652–691.CrossRefGoogle Scholar
[9]Kijima, M., “Quasi-limiting distributions of Markov chains that are skip-free to the left in continuous-time”, J. Appl. Probab. 30 (1993) 509–517.CrossRefGoogle Scholar
[10]Kijima, M., Nair, M. G., Pollett, P. K. and van Doom, E., “Limiting conditional distributions for birth-death processes”, Adv. Appl. Probab. 29 (1997) 185–204.CrossRefGoogle Scholar
[11]Kurtz, T. G., “Solutions of ordinary differential equations as limits of pure jump Markov processes”, J. Appl. Probab. 7 (1970) 49–58.CrossRefGoogle Scholar
[12]Kurtz, T. G., “Limit theorems for sequences of jump Markov processes approximating ordinary differential processes”, J. Appl. Probab. 8 (1971) 344–356.CrossRefGoogle Scholar
[13]Kurtz, T. G., “Limit theorems and diffusion approximations for density dependent Markov chains”, Math. Prog. Study 5 (1976) 67–78.CrossRefGoogle Scholar
[14]Mandl, P., “On the asymptotic behaviour of probabilities within groups of states of a homogeneous Markov processes” (in Czech), Časopis Pěst. Mat. 85 (1960) 448–456.CrossRefGoogle Scholar
[15]Meyer, P. A., Smithe, R. T. and Walsh, J. B., “Birth and death of Markov processes”, in Proc. 6th Berkeley Symp. Math. Statist. Probab. 3, (1971), 295–305.Google Scholar
[16]Nair, M. G. and Pollett, P. K., “On the relationship between μ-invariant measures and quasistationary distributions for continuous-time Markov chains”, Adv. Appl. Probab. 25 (1993) 82–102.CrossRefGoogle Scholar
[17]Pollett, P. K., “On the equivalence of μ-invariant measures for the minimal process and its q−matrix”, Stochastic Process. Appl. 22 (1986) 203–221.CrossRefGoogle Scholar
[18]Pollett, P. K., “Reversibility, invariance and μ-invariance”, Adv. Appl. Probab. 20 (1988) 600–621.CrossRefGoogle Scholar
[19]Pollett, P. K. and Vere-Jones, D., “A note on evanescent processes”, Austral. J. Statist. 34 (1992) 531–536.CrossRefGoogle Scholar
[20]Tweedie, R. L., “Some ergodic properties of the Feller minimal process”, Quart. J. Math. Oxford 25(1974) 485–495.CrossRefGoogle Scholar
[21]Tweedie, R. L., “Truncation approximations of invariant measures for Markov chains”, J. Appl. Probab. 35 (1999) 517–536.CrossRefGoogle Scholar
[22]van Doom, E. A., “Quasi-stationary distributions and convergence to quasi-stationarity of birthdeath processes”, Adv. Appl. Probab. 23 (1991) 683–700.CrossRefGoogle Scholar
[23]van Doom, E. A. and Schrijner, P., “Limit theorems for discrete-time Markov chains on the nonnegative integers conditioned on recurrence to zero”, Stochastic Models 14 (1996) 77–102.CrossRefGoogle Scholar
[24]Vere-Jones, D., “Some limit theorems for evanescent processes”, Austral. J. Statist. 11 (1969) 67–78.CrossRefGoogle Scholar