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Some results for quasi-stationary distributions of birth-death processes

Published online by Cambridge University Press:  14 July 2016

Masaaki Kijima  and
E. Seneta
Masaaki Kijima*
Affiliation:
The University of Tsukuba, Tokyo
E. Seneta*
Affiliation:
University of Sydney
*
∗ Postal address: Graduate School of Systems Management, The University of Tsukuba, Tokyo, Otsuka, Bunkyo-ku, Tokyo 112, Japan.
∗∗ Postal address: Department of Mathematical Statistics, University of Sydney, Sydney, NSW 2006, Australia.
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Abstract

Quasi-stationary distributions are considered in their own right, and from the standpoint of finite approximations, for absorbing birth-death processes. Results on convergence of finite quasi-stationary distributions and a stochastic bound for an infinite quasi-stationary distribution are obtained. These results are akin to those of Keilson and Ramaswamy (1984). The methodology is a synthesis of Good (1968) and Cavender (1978).

Keywords

INVARIANT MEASURESORTHOGONAL POLYNOMIALSSTOCHASTIC BOUNDSTRUNCATIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 3 , September 1991 , pp. 503 - 511
Copyright
Copyright © Applied Probability Trust 1991 

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References

Cavender, J. A. (1978) Quasi-stationary distributions of birth-and-death processes. Adv. Appl. Prob. 10, 570–586.Google Scholar
Darroch, J. N. and Seneta, E. (1965) On quasi-stationary distributions in absorbing discrete finite Markov chains. J. Appl. Prob. 2, 88–100.Google Scholar
Darroch, J. N. and Seneta, E. (1967) On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Prob. 4, 192–196.Google Scholar
Good, P. (1968) The limiting behavior of transient birth and death processes conditioned on survival. J. Austral. Math. Soc. 8, 716–722.Google 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.Google Scholar
Karlin, S. and Mcgregor, J. L. (1957b) The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.CrossRefGoogle Scholar
Keilson, J. and Ramaswamy, R. (1984) Convergence of quasi-stationary distributions in birth-death processes. Stoch. Proc. Appl. 24, 631–643.Google Scholar
Kesten, H. (1970) Review of Good (1968). Math. Review 39, 410.Google Scholar
Mandl, P. (1960) On the asymptotic behaviour of probabilities of groups of states of a homogeneous Markov process. Cas. Pest. Mat. 85, 448–456 (in Russian).Google Scholar
Ledermann, W. and Reuter, G. E. H. (1954) Spectral theory for the differential equations of simple birth and death processes. Phil. Trans. R. Soc. London A 246, 321–369.Google Scholar
Pollett, P. K. (1988) Reversibility, invariance and µ-invariance. Adv. Appl. Prob. 20, 600–621.Google Scholar
Pollett, P. K. (1989) The generalized Kolmogorov criterion. Stoch. Proc. Appl. 33, 29–44.Google Scholar
Seneta, E. (1966) Quasi-stationary behaviour in the random walk with continuous time. Austral. J. Statist. 8, 92–98.Google 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. Prob. 3, 403–434.Google Scholar
Szegö, G. (1959) Orthogonal Polynomials. American Math. Society Colloq. Publ. 23, American Math. Soc., New York.Google Scholar
Van Doorn, E. (1985) Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process. Adv. Appl. Prob. 17, 514–530.CrossRefGoogle Scholar
Van Doorn, E. (1987) Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices. J. Approx. Theory 51, 254–266.Google Scholar
Van Doorn, E. (1990) Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Unpublished.Google Scholar
Vere-Jones, D. (1969) Some limit theorems for evanescent processes. Austral. J. Statist. 11, 67–78.Google Scholar