Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-26T09:59:02.184Z Has data issue: false hasContentIssue false
  • English
  • Français

Additive Functionals for Discrete-Time Markov Chains with Applications to Birth-Death Processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Yuanyuan Liu
Yuanyuan Liu*
Affiliation:
Central South University
*
∗ Postal address: School of Mathematics, Railway Campus, Central South University, Changsha, Hunan, 410075, China. Email address: liuyy@csu.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we are interested in bounding or calculating the additive functionals of the first return time on a set for discrete-time Markov chains on a countable state space, which is motivated by investigating ergodic theory and central limit theorems. To do so, we introduce the theory of the minimal nonnegative solution. This theory combined with some other techniques is proved useful for investigating the additive functionals. This method is used to study the functionals for discrete-time birth-death processes, and the polynomial convergence and a central limit theorem are derived.

Keywords

Additive functionalbirth-death processergodicitycentral limit theorem

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J55: Local time and additive functionals
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 925 - 937
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Asmussen, S. and Bladt, M. (1994). Poisson's equation for queues driven by a Markovian marked point process. Queueing Systems 17, 235–274.Google Scholar
[2] Chen, M.-F. (2003). From Markov Chains to Non-Equilibrium Particle Systems, 2nd edn. World Scientific, Singapore.Google Scholar
[3] Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer, New York.Google Scholar
[4] Douc, R., Fort, G., Moulines, E. and Soulier, P. (2004). Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Prob. 14, 1353–1377.Google Scholar
[5] Fort, G. and Moulines, E. (2003). Polynomial ergodicity of Markov transition kernels. Stoch. Process. Appl. 103, 57–99.Google Scholar
[6] Glynn, P. W. (1994). Poisson's equation for the recurrent M/G/1 queue. Adv. Appl. Prob. 26, 1044–1062.Google Scholar
[7] Grassmann, W. K. (1987). The asymptotic variance of a time average in a birth-death process. Ann. Operat. Res. 8, 165–174.Google Scholar
[8] Hou, Z. T. and Guo, Q. F. (1988). Homogeneous Denumerable Markov Processes. Springer, Berlin.Google Scholar
[9] Jarner, S. F. and Roberts, G. O. (2002). Polynomial convergence rates of Markov chains. Ann. Appl. Prob. 12, 224–247.Google Scholar
[10] Jones, G. L. (2004). On the Markov chain central limit theorem. Prob. Surveys 1, 299–320.Google Scholar
[11] Liu, Y. (2010). Augmented truncation of discrete-time Markov chains. Operat. Res. Lett. 38, 218–222.Google Scholar
[12] Liu, Y., Zhang, H. and Zhao, Y. (2010). Subgeometric ergodicity for continuous-time Markov chains. J. Math. Anal. Appl. 368, 178–189.Google Scholar
[13] Mao, Y. (2002). Ergodicity for discrete-time random walks. Beijing Shifan Daxue Xuebao 38, 729–733.Google Scholar
[14] Mao, Y. (2003). Algebraic convergence for discrete-time ergodic Markov chains. Sci. China Ser. A 46, 621–630.Google Scholar
[15] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
[16] Nummelin, E. and Tuominen, P. (1982). Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory. Stoch. Process. Appl. 12, 187–202.Google Scholar
[17] Roberts, G. O. and Rosenthal, J. S. (2004). General state space Markov chains and MCMC algorithms. Prob. Surveys 1, 20–71.Google Scholar
[18] Tuominen, P. and Tweedie, R. L. (1994). “Subgeometric rates of convergence of f-ergodic Markov chains. Adv. Appl. Prob. 26, 775–798.Google Scholar
[19] Tweedie, R. L. (1983). Criteria for rates of convergence of Markov chains. In Probability Statistics and Analysis (London Math. Soc. Lecture Notes Ser. 79), eds Kingman, J. F. C. and E, G. Reuter, H., Cambridge University Press, pp. 260–276.Google Scholar
[20] Van Doorn, E. A. and Schrijner, P. (1995). Geometric ergodicity and quasi-stationarity in discrete-time birth-death processes. J. Austral. Math. Soc. 37, 121–144.Google Scholar
[21] Whitt, W. (1992). Asymptotic formulas for Markov processes with applications to simulations. Operat. Res. 40, 279–291.Google Scholar