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Computable Bounds for the Decay Parameter of a Birth–Death Process

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

David Sirl ,
Hanjun Zhang  and
Phil Pollett
David Sirl*
Affiliation:
The University of Queensland
Hanjun Zhang*
Affiliation:
The University of Queensland
Phil Pollett*
Affiliation:
The University of Queensland
*
∗ Postal address: Department of Mathematics, The University of Queensland, Brisbane, QLD 4072, Australia.
∗ Postal address: Department of Mathematics, The University of Queensland, Brisbane, QLD 4072, Australia.
∗ Postal address: Department of Mathematics, The University of Queensland, Brisbane, QLD 4072, Australia.
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Abstract

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We present bounds on the decay parameter for absorbing birth–death processes adapted from results of Chen (2000), (2001). We address numerical issues associated with computing these bounds, and assess their accuracy for several models, including the stochastic logistic model, for which estimates of the decay parameter have been obtained previously by Nåsell (2001).

Keywords

Approximation; exponential ergodicityquasi-stationary distribution

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 65C20: Models, numerical methods
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 2 , June 2007 , pp. 476 - 491
Copyright
Copyright © Applied Probability Trust 2007 

References

Anderson, W. J. (1991). Continuous-Time Markov Chains: An Applications Oriented Approach. Springer, New York.Google Scholar
Bordes, G. and Roehner, B. (1983). Application of Stieltjes theory for S-fractions to birth and death processes. Adv. Appl. Prob. 15, 507–530.Google Scholar
Breyer, L. A. and Hart, A. G. (2000). Approximations of quasi-stationary distributions for Markov chains. Math. Comput. Modelling 31, 69–79.Google Scholar
Callaert, H. and Keilson, J. (1973). On exponential ergodicity and spectral structure for birth–death processes. I. Stoch. Process. Appl. 1, 187–216.Google Scholar
Callaert, H. and Keilson, J. (1973). On exponential ergodicity and spectral structure for birth–death processes. II. Stoch. Process. Appl. 1, 217–235.Google Scholar
Chen, M.-F. (1991). Exponential L2-convergence and L2-spectral gap for Markov processes. Acta. Math. Sinica New Ser. 7, 19–37.Google Scholar
Chen, M.-F. (1996). Estimation of spectral gap for Markov chains. Acta. Math. Sinica New Ser. 12, 337–360.Google Scholar
Chen, M.-F. (2000). Explicit bounds of the first eigenvalue. Sci. China Ser. A 43, 1051–1059.Google Scholar
Chen, M.-F. (2001). Variational formulas and approximation theorems for the first eigenvalue in dimension one. Sci. China Ser. A 44, 409–418.Google Scholar
Chen, M.-F. (2005). Eigenvalues, Inequalities, and Ergodic Theory. Springer, London.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
Kartashov, N. V. (2000). Calculation of the spectral ergodicity exponent for the birth and death process. Ukrainian Math. J. 52, 1018–1028.Google Scholar
Kijima, M. (1992). Evaluation of the decay parameter for some specialized birth–death processes. J. Appl. Prob. 29, 781–791.Google Scholar
Kingman, J. F. C. (1963). The exponential decay of Markov transition probabilities. Proc. London Math. Soc. (3) 13, 337–358.Google Scholar
Mao, Y.-H. (2006). Convergence rates in strong ergodicity for Markov processes. Stoch. Process. Appl. 116, 1964–1964.Google Scholar
Nåsell, I. (1996). The quasi-stationary distribution of the closed endemic SIS model. Adv. Appl. Prob. 28, 895–932.Google Scholar
Nåsell, I. (1999). On the quasi-stationary distribution of the stochastic logistic epidemic. Math. Biosci. 156, 21–40.Google Scholar
Nåsell, I. (2001). Extinction and quasi-stationarity in the Verhulst logistic model. J. Theoret. Biol. 211, 11–27.Google Scholar
Pollett, P. K. (1995). The determination of quasistationary distributions directly from the transition rates of an absorbing Markov chain. Math. Comput. Modelling 22, 279–287.Google Scholar
Reuter, G. E. H. (1957). Denumerable Markov processes and the associated contraction semigroups on l. Acta Math. 97, 1–46.Google Scholar
Roehner, B. and Valent, G. (1982). Solving the birth and death processes with quadratic asymptotically symmetric transition rates. SIAM J. Appl. Math. 42, 1020–1046.Google Scholar
Seneta, E. (1966). Quasi-stationary behaviour in the random walk with continuous time. Austral. J. Statist. 8, 92–98.Google Scholar
Tweedie, R. L. (1981). Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes. J. Appl. Prob. 18, 122–130.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. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth–death processes. Adv. Appl. Prob. 23, 683–700.Google Scholar
Van Doorn, E. A. (2002). Representations for the rate of convergence of birth–death processes. Theory Prob. Math. Statist. 65, 37–43.Google Scholar
Waugh, W. A. O'N. (1958). Conditioned Markov processes. Biometrika 45, 241–249.Google Scholar
Ze{ı˘fman, A. I.} (1991). Some estimates of the rate of convergence for birth and death processes. J. Appl. Prob. 28, 268–277.Google Scholar
Zhang, H., Chen, A., Lin, X. and Hou, Z. (2001). Strong ergodicity of monotone transition functions. Statist. Prob. Lett. 55, 63–69.CrossRefGoogle Scholar