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Stability and exponential convergence of continuous-time Markov chains

  • A. Yu. Mitrophanov (a1)


For finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, we derive perturbation bounds which demonstrate the connection between the sensitivity to perturbations and the rate of exponential convergence to stationarity. Our perturbation bounds substantially improve upon the known results. We also discuss convergence bounds for chains with diagonalizable generators and investigate the relationship between the rate of convergence and the sensitivity of the eigenvalues of the generator; special attention is given to reversible chains.


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Postal address: Faculty of Computer Science and Information Technology, Saratov State University, 83 Astrakhanskaya str., Saratov 410012, Russia. Email address:


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[1] Andreev, D. B. et al. (2002). On ergodicity and stability estimates for some nonhomogeneous Markov chains. J. Math. Sci. (New York) 112, 41114118.
[2] Arakelian, V. B., Wild, J. R., and Simonian, A. L. (1998). Investigation of stochastic fluctuations in the signal formation of microbiosensors. Biosensors Bioelectron. 13, 5559.
[3] Ball, F. G., Milne, R. K., and Yeo, G. F. (2000). Stochastic models for systems of interacting ion channels. IMA J. Math. Appl. Med. Biol. 17, 263293.
[4] Diaconis, P., and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Prob. 6, 695750.
[5] Diaconis, P., and Saloff-Coste, L. (1996). Nash inequalities for finite Markov chains. J. Theoret. Prob. 9, 459510.
[6] Diaconis, P., and Strook, D. (1991). Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Prob. 1, 3661.
[7] Fill, J. A. (1991). Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process. Ann. Appl. Prob. 1, 6287.
[8] Granovsky, B. L., and Zeifman, A. I. (1997). The decay function of nonhomogeneous birth—death processes, with application to mean-field models. Stoch. Process. Appl. 72, 105120.
[9] Granovsky, B. L., and Zeifman, A. I. (2000). Nonstationary Markovian queues. J. Math. Sci. (New York) 99, 14151438.
[10] Granovsky, B. L., and Zeifman, A. I. (2000). The N-limit of spectral gap of a class of birth—death Markov chains. Appl. Stoch. Models Business Industry 16, 235248.
[11] Horn, R. A., and Johnson, C. R. (1985). Matrix Analysis. Cambridge University Press.
[12] Kartashov, N. V. (1986). Inequalities in theorems of ergodicity and stability for Markov chains with common state space. I. Theory Prob. Appl. 30, 247259.
[13] Kartashov, N. V. (1986). Inequalities in theorems of ergodicity and stability for Markov chains with common state space. II. Theory Prob. Appl. 30, 507515.
[14] Peng, N. F. (1996). Spectral representations of the transition probability matrices for continuous time finite Markov chains. J. Appl. Prob. 33, 2833.
[15] Seneta, E. (1979). Coefficients of ergodicity: structure and applications. Adv. Appl. Prob. 11, 576590.
[16] Thoumine, O., and Meister, J.-J. (2000). A probabilistic model for ligand-cytoskeleton transmembrane adhesion: predicting the behaviour of microspheres on the surface of migrating cells. J. Theoret. Biol. 204, 381392.
[17] Van Kampen, N. G. (1981). Stochastic Processes in Physics and Chemistry. North Holland, Amsterdam.
[18] Zeifman, A. I. (1985). Stability for continuous-time nonhomogeneous Markov chains. In Stability Problems for Stochastic Models (Lecture Notes Math. 1155), eds Kalashnikov, V. V. and Zolotarev, V. M., Springer, Berlin, pp. 401-414.
[19] Zeifman, A. I. (1995). Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes. Stoch. Process. Appl. 59, 157173.
[20] Zeifman, A. I. (1998). Stability of birth-and-death processes. J. Math. Sci. (New York) 91, 30233031.
[21] Zeifman, A. I., and Isaacson, D. L. (1994). On strong ergodicity for nonhomogeneous continuous-time Markov chains. Stoch. Process. Appl. 50, 263273.
[22] Zheng, Q. (1998). Note on the non-homogeneous Prendiville process. Math. Biosci. 148, 15.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
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