On quasi-stationary distributions in absorbing continuous-time finite Markov chains

J. N. Darroch; E. Seneta

doi:10.2307/3212311

Extract

In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.

References

[1] Darroch, J. N. and Seneta, E. (1965) On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Prob. 2, 88–100.Google Scholar

[2] Doob, J. L. (1953) Stochastic Processes. John Wiley, New York.Google Scholar

[3] Kemeny, J. G. and Snell, L. J. (1961) Finite continuous time Markov chains. Teor. Veroyat. Primen. 6, 110–115.Google Scholar

[4] Mandl, P. (1960) On the asymptotic behaviour of probabilities within groups of states of a homogeneous Markov process. Cas. pest. mat. 85, 448–456 (in Russian).Google Scholar

[5] Seneta, E. (1966) Quasi-stationary behaviour in the random walk with continuous time. Aust. J. Statist. 8, 92–98.Google Scholar

[6] Seneta, E. (1966) Quasi-stationary distributions and time-reversion in genetics. J. R. Statist. Soc. B 28, 253–277.Google Scholar

[7] Seneta, E. and Vere-Jones, D. (1966) On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434.Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Kyprianou, E. K. 1971. On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals. Journal of Applied Probability, Vol. 8, Issue. 03, p. 494.

Iosifescu, M. and Tăutu, P. 1973. Stochastic processes and applications in biology and medicine I. Vol. 3, Issue. , p. 163.

Keilson, J. 1974. Sojourn times, fluctuation and rare events; the role of jitter. Advances in Applied Probability, Vol. 6, Issue. 2, p. 197.

Flaspohler, David C. 1974. Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains. Annals of the Institute of Statistical Mathematics, Vol. 26, Issue. 1, p. 351.

Ivan, Constantin 1981. Weak convergence of the simple birth-and-death process. Journal of Applied Probability, Vol. 18, Issue. 01, p. 245.

Kalpakam, S. and Hameed, M. A. Shahul 1983. Quasi-stationary distribution of a two-unit warm-standby redundant system. Journal of Applied Probability, Vol. 20, Issue. 02, p. 429.

Keilson, Julian and Ramaswamy, Ravi 1984. Convergence of quasi-stationary distributions in birth-death processes. Stochastic Processes and their Applications, Vol. 18, Issue. 2, p. 301.

Keilson, Julian and Ramaswamy, Ravi 1986. The bivariate maximum process and quasi-stationary structure of birth-death processes. Stochastic Processes and their Applications, Vol. 22, Issue. 1, p. 27.

Pollett, P.K. 1986. On the equivalence of μ-invariant measures for the minimal process and its q-matrix. Stochastic Processes and their Applications, Vol. 22, Issue. 2, p. 203.

Parsons, R. W. and Pollett, P. K. 1987. Quasistationary distributions for autocatalytic reactions. Journal of Statistical Physics, Vol. 46, Issue. 1-2, p. 249.

Keilson, Julian and Ramaswamy, Ravi 1987. The Relaxation time for truncated birth-death processes. Probability in the Engineering and Informational Sciences, Vol. 1, Issue. 4, p. 367.

Ziedins, Ilze 1987. Quasi-stationary distributions and one-dimensional circuit-switched networks. Journal of Applied Probability, Vol. 24, Issue. 4, p. 965.

Pollett, P. K. 1988. Reversibility, invariance and μ-invariance. Advances in Applied Probability, Vol. 20, Issue. 3, p. 600.

Pollett, P. K. 1988. On the problem of evaluating quasistationary distributions for open reaction schemes. Journal of Statistical Physics, Vol. 53, Issue. 5-6, p. 1207.

Pollett, P. K. and Roberts, A. J. 1990. A description of the long-term behaviour of absorbing continuous-time Markov chains using a centre manifold. Advances in Applied Probability, Vol. 22, Issue. 1, p. 111.

Evans, Suzanne P. 1990. A quasi-stationary analysis of a virtual path in a B-ISDN network shared by services with very different characteristics. Computer Networks and ISDN Systems, Vol. 20, Issue. 1-5, p. 391.

Gerontidis, Ioannis I. 1990. ON THE VARIANCE‐COVARIANCE MATRIX IN MARKOVIAN MANPOWER SYSTEMS IN CONTINUOUS TIME. Australian Journal of Statistics, Vol. 32, Issue. 3, p. 271.

Kijima, Masaaki and Seneta, E. 1991. Some results for quasi-stationary distributions of birth-death processes. Journal of Applied Probability, Vol. 28, Issue. 03, p. 503.

Kijima, Masaaki 1992. On the existence of quasi-stationary distributions in denumerable R-transient Markov chains. Journal of Applied Probability, Vol. 29, Issue. 01, p. 21.

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On quasi-stationary distributions in absorbing continuous-time finite Markov chains

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