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COMPUTABLE STRONGLY ERGODIC RATES OF CONVERGENCE FOR CONTINUOUS-TIME MARKOV CHAINS

Part of: Markov processes

Published online by Cambridge University Press:  01 April 2008

YUANYUAN LIU ,
HANJUN ZHANG  and
YIQIANG ZHAO
YUANYUAN LIU*
Affiliation:
School of Mathematics, Railway Campus, Central South University, Changsha, Hunan, 410075, PR China (email: liuyy@csu.edu.cn)
HANJUN ZHANG
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, PR China (email: hjzhang001@gmail.com)
YIQIANG ZHAO
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S 5B6, Canada (email: zhao@math.carleton.ca)
*
✉For correspondence; e-mail: liuyy@csu.edu.cn
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Abstract

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In this paper, we investigate computable lower bounds for the best strongly ergodic rate of convergence of the transient probability distribution to the stationary distribution for stochastically monotone continuous-time Markov chains and reversible continuous-time Markov chains, using a drift function and the expectation of the first hitting time on some state. We apply these results to birth–death processes, branching processes and population processes.

Keywords

strong ergodicityconvergence ratebirth and death processbranching processpopulation process

MSC classification

Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
The ANZIAM Journal , Volume 49 , Issue 4 , April 2008 , pp. 463 - 478
Copyright
Copyright © Australian Mathematical Society 2008

References

[1]Anderson, W. J., Continuous-time Markov chains. An applications-oriented approach (Springer-Verlag, New York, 1991).Google Scholar
[2]Baxendal, P. H., “Renewal theory and computable convergence rates for geometrically ergodic Markov chains”, Ann. Appl. Probab. 15 (2005) 700–738.Google Scholar
[3]Chen, A. Y., “Ergodicity and stability generalized Markov branching processes with resurrection”, J. Appl. Probab. 39 (2002) 786–803.CrossRefGoogle Scholar
[4]Chen, M. F., “Estimation of spectral gap for Markov chains”, Acta Math. Sin. New Ser. 12 (1996) 337–360.Google Scholar
[5]Chen, M. F., “Equivelence of exponential ergodicity and L 2-exponential convergence for Markov chains”, Stochastic Process Appl. 87 (2000) 281–297.CrossRefGoogle Scholar
[6]Chen, R. R., “An extended class of time-continuous branching processes”, J. Appl. Probab. 34 (1997) 14–23.CrossRefGoogle Scholar
[7]Diaconis, P. and Saloff-Coste, L., “Nash’s inequality for finite Markov chains”, J. Theor. Probab. 9 (1996) 459–510.CrossRefGoogle Scholar
[8]Doeblin, W., “Sur les propriétés asymptotiquea du mouvement régis par certain types de chaine simples”, Bull. Math. Soc. Roum. Sci. 39 (1937) 57–115.Google Scholar
[9]Down, D., Meyn, S. P. and Tweedie, R. L., “Exponential and uniform ergodicity of Markov processes”, Ann. Probab. 23 (1995) 1671–1691.CrossRefGoogle Scholar
[10]Hou, Z. T. and Guo, Q. F., Homogeneous denumerable Markov processes (Springer, New York, 1988).Google Scholar
[11]Isaacson, D. and Luecke, G. R., “Strongly ergodic Markov chains and rates of convergence using spectral conditions”, Stochastic Process Appl. 7 (1978) 113–121.CrossRefGoogle Scholar
[12]Isaacson, D. and Tweedie, R. L., “Criteria for strong ergodicity for Markov chains”, J. Appl. Probab. 15 (1978) 87–95.CrossRefGoogle Scholar
[13]Kingman, J. F. C., “Markov population processes”, J. Appl. Probab. 6 (1969) 1–18.CrossRefGoogle Scholar
[14]Lund, R. B., Meyn, S. P. and Tweedie, R. L., “Computable exponential convergence rates for stochastically ordered Markov processes”, Ann. Appl. Probab. 6 (1996) 218–237.CrossRefGoogle Scholar
[15]Lund, R. B. and Tweedie, R. L., “Geometric convergence rates for stochastically ordered Markov chains”, Math. Oper. Res. 20 (1996) 182–194.CrossRefGoogle Scholar
[16]Mao, Y. H., “Strong ergodicity for Markov processes by coupling methods”, J. Appl. Probab. 39 (2002) 839–852.CrossRefGoogle Scholar
[17]Mao, Y. H., “Convergence rates in strong ergodicity for Markov processes”, Stochastic Process Appl. 116 (2006) 1964–1976.CrossRefGoogle Scholar
[18]Reuter, G. E. H., “Competition processes”, Proc. Fourth Berkeley Symp. Math. Statist. Probab. 2 (1961) 421–430.Google Scholar
[19]Tweedie, R. L., “Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes”, J. Appl. Probab. 18 (1981) 122–130.CrossRefGoogle Scholar
[20]Zhang, H. J. and Chen, A. Y., “Stochastic comparability and dual q-functions”, J. Math. Anal. Appl. 234 (1999) 482–499.CrossRefGoogle Scholar
[21]Zhang, H. J., Chen, A. Y., Lin, X. and Hou, Z. T., “Strong ergodicity of monotone transition functions”, Statist. Probab. Lett. 55 (2001) 63–69.CrossRefGoogle Scholar
[22]Zhang, Y. H., “Strong ergodicity for single-birth processes”, J. Appl. Probab. 38 (2001) 270–277.CrossRefGoogle Scholar