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Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes

Published online by Cambridge University Press:  01 July 2016

Erik A. Van Doorn
Erik A. Van Doorn*
Affiliation:
University of Twente
*
∗Postal address: Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands.
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Abstract

For a birth–death process (X(t), ) on the state space {−1, 0, 1, ·· ·}, where −1 is an absorbing state which is reached with certainty and {0, 1, ·· ·} is an irreducible class, we address and solve three problems. First, we determine the set of quasi-stationary distributions of the process, that is, the set of initial distributions which are such that the distribution of X(t), conditioned on non-absorption up to time t, is independent of t. Secondly, we determine the quasi-limiting distribution of X(t), that is, the limit as t→∞ of the distribution of X(t), conditioned on non-absorption up to time t, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of X(t), conditioned on non-absorption up to time t, to their limits. Some examples conclude the paper. Our main tools are the spectral representation for the transition probabilities of a birth–death process and a duality concept for birth–death processes.

Keywords

DECAY PARAMETERDUALITYORTHOGONAL POLYNOMIALSQUASI-LIMITINGDISTRIBUTIONQUASI-STATIONARY DISTRIBUTIONRATE OF CONVERGENCESPECTRAL REPRESENTATION
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 4 , December 1991 , pp. 683 - 700
Copyright
Copyright © Applied Probability Trust 1991 

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References

Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions. National Bureau of Standards Applied Mathematics Series 55, Washington, DC.Google Scholar
Buiculescu, M. (1972) Quasi-stationary distributions for continuous-time Markov processes with a denumerable set of states. Rev. Roumaine Math. Pures Appl. 17, 1013–1023.Google Scholar
Cavender, J. A. (1978) Quasi-stationary distributions of birth-and-death processes. Adv. Appl. Prob. 10, 570–586.CrossRefGoogle Scholar
Dambrine, S. and Moreau, M. (1981) Note on the stochastic theory of a self-catalytic chemical reaction, I. Physica 106A, 559–573.CrossRefGoogle Scholar
Van Doorn, E. A. (1984) On oscillation properties and the interval of orthogonality of orthogonal polynomials. SIAM J. Math. Anal. 15, 1031–1042.Google Scholar
Van Doorn, E. A. (1985) Conditions for exponential ergodicity and bounds for the decay parameter of a birth–death process. Adv. Appl. Prob. 17, 514–530.Google Scholar
Van Doorn, E. A. (1986) On orthogonal polynomials with positive zeros and the associated kernel polynomials. J. Math. Anal. Appl. 113, 441–450.Google Scholar
Van Doorn, E. A. (1987) Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices. J. Approx. Th. 51, 254–266.Google Scholar
Flaspohler, D. C. (1974) Quasi-stationary distributions in absorbing continuous-time denumerable Markov chains. Ann. Inst. Statist. Math. 26, 351–356.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
Ismail, M. E. H. (1987) The variation of zeros of certain orthogonal polynomials. Adv. Appl. Math. 8, 111–118.Google Scholar
Ismail, M. E. H., Letessier, J. and Valent, G. (1988). Linear birth and death models and associated Laguerre and Meixner polynomials. J. Approx. Th. 55, 337–348.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.CrossRefGoogle Scholar
Karlin, S. and Mcgregor, J. L. (1957b) The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.Google Scholar
Karlin, S. and Mcgregor, J. L. (1958a) Many server qeueing processes with Poisson input and exponential service times. Pacific J. Math. 8, 87–118.CrossRefGoogle Scholar
Karlin, S. and Mcgregor, J. L. (1958b) Linear growth, birth and death processes. J. Math. Mech. (now Indiana Univ. Math. J.) 7, 643–662.Google Scholar
Kemperman, J. H. B. (1962) An analytical approach to the differential equations of the birth-and-death process. Michigan Math. J. 9, 321–361.Google Scholar
Kesten, H. (1970) Review of Good (1968). Math. Rev. 39, 410.Google Scholar
Kijima, M. and Seneta, E. (1991) Some results for quasi-stationary distributions of birth-death processes. J. Appl. Prob. 28(3).Google Scholar
Kingman, J. F. C. (1963) The exponential decay of Markov transition probabilities. Proc. London Math. Soc. (3) 13, 337–358.CrossRefGoogle Scholar
Parsons, R. W. and Pollett, P. K. (1987) Quasistationary distributions for autocatalytic reactions. J. Statist. Phys. 46, 249–254.CrossRefGoogle 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.CrossRefGoogle Scholar
Pollett, P. K. (1989a) On the problem of evaluating quasi-stationary distributions for open reaction schemes. J. Statist. Phys. 53, 1207–1215.Google Scholar
Pollett, P. K. (1989b) The generalized Kolomogorov criterion. Stoch. Proc. Appl. 33, 29–44.Google Scholar
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. Adv. Appl. Prob. 22, 111–128.Google Scholar
Reuter, G. E. H. (1957) Denumerable Markov processes and the associated contraction semi-groups on l . Acta Math. 97, 1–46.Google Scholar
Reuter, G. E. H. and Ledermann, W. (1953) On the differential equations for the transition probabilities of Markov processes with enumerably many states. Proc. Camb. Phil. Soc. 49, 247–262.CrossRefGoogle Scholar
Seneta, E. (1966) Quasi-stationary behaviour in the random walk with continuous time. Austral. J. Statist. 8, 92–98.Google Scholar
Vere-Jones, D. (1969) Some limit theorems for evanescent processes. Austral. J. Statist. 11, 67–78.Google Scholar
Whitt, W. (1985) The renewal-process stationary-excess operator. J. Appl. Prob. 22, 156–167.Google Scholar