Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-25T06:31:19.208Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-ergodicity for non-homogeneous continuous-time Markov chains

Published online by Cambridge University Press:  14 July 2016

A. I. Zeifman
A. I. Zeifman*
Affiliation:
Vologda State Pedagogical Institute
*
∗Postal address: Vologda State Pedagogical Institute, S. Orlova 6, 160600 Vologda, USSR.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a non-homogeneous continuous-time Markov chain X(t) with countable state space. Definitions of uniform and strong quasi-ergodicity are introduced. The forward Kolmogorov system for X(t) is considered as a differential equation in the space of sequences l1. Sufficient conditions for uniform quasi-ergodicity are deduced from this equation. We consider conditions of uniform and strong ergodicity in the case of proportional intensities.

Keywords

FORWARD KOLMOGOROV SYSTEMDIFFERENTIAL EQUATION IN l1 SPACEERGODICITYBIRTH-AND-DEATH PROCESS
Type
Short Communications
Information
Journal of Applied Probability , Volume 26 , Issue 3 , September 1989 , pp. 643 - 648
Copyright
Copyright © Applied Probability Trust 1989 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bharucha-Reid, A. T. (1960) Elements of the Theory of Markov Processes and Their Applications. McGraw-Hill, New York.Google Scholar
[2] Bohme, O. (1982) Periodic Markov transition functions 1, 2. Math. Nachr. 108, 231–239; 109, 47–56.CrossRefGoogle Scholar
[3] Daletzky, Yu.L. and Krein, M. G. (1970) Stability of Solutions of Differential Equations in Banach Space. Nauka, Moscow (in Russian).Google Scholar
[4] Gnedenko, B. V. and Makarov, I. P. (1971) Properties of solutions of the loss periodic intensities problem. Diff. Eqns 7, 1696–1698 (in Russian).Google Scholar
[5] Goel, N. and Richter-Dyn, N. (1974) Stochastic Models in Biology. Academic Press, New York.Google Scholar
[6] Griffeath, D. (1975) Uniform coupling of nonhomogeneous Markov chains. J. Appl. Prob. 12, 753–763.CrossRefGoogle Scholar
[7] Hartman, P. (1964) Ordinary Differential Equations. Wiley, New York.Google Scholar
[8] Isaacson, D. and Arnold, B. (1978) Strong ergodicity for continuous-time Markov chains. J. Appl. Prob. 15, 699–706.CrossRefGoogle Scholar
[9] Isaacson, D. and Seneta, E. (1982) Ergodicity for countable inhomogeneous Markov chains. Linear Algebra Appl. 48, 37–44.CrossRefGoogle Scholar
[10] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, Chichester.Google Scholar
[11] Kovalenko, I. N. and Sarmanov, ?. V. (1978) Brief Course in Stochastic Process Theory. Visha Shkola, Kiev (in Russian).Google Scholar
[12] Oguztoreli, M. N. (1972) On an infinite system of differential equations occurring in the degradations of polymers. I. Util. Math. 1, 141–155.Google Scholar
[13] Scott, M., Arnold, B. and Isaacson, D. (1982) Strong ergodicity for continuous-time, nonhomogeneous Markov chains. J. Appl. Prob. 19, 692–694.CrossRefGoogle Scholar
[14] Scott, ?. and Isaacson, D. (1983) Proportional intensities and strong ergodicity for Markov processes. J. Appl. Prob. 20, 185–190.CrossRefGoogle Scholar
[15] Seneta, E. (1981) Non-negative Matrices and Markov Chains. Springer-Verlag, New York.CrossRefGoogle Scholar
[16] Shahbazov, A. A. (1982) On a class of Markov processes with varying intensities and their applications to queueing theory. Optimization 13, 133–144 (in Russian).Google Scholar
[17] Vassiliou, P. C. (1982) Asymptotic behaviour of Markov systems. J. Appl. Prob. 19, 851–857.CrossRefGoogle Scholar
[18] Yoon, Y. J. (1977) Stability of an infinite system of differential equations for the kinetics of polymer degradation. Dyn. Syst. Proc. Univ. Fla. Int. Symp., Gainesville, 1976, New York, 507–511.CrossRefGoogle Scholar
[19] Zeifman, A. I. (1983) On asymptotic behaviour of solutions of the forward Kolmogorov system. Ukr. Math. J. 35, 621–624 (in Russian).Google Scholar
[20] Zeifman, A. I. (1985) Asymptotic behaviour of the mean of correct birth-and-death processes. Ukr. Math. J. 37, 253–256 (in Russian).Google Scholar
[21] Zeifman, A. I. (1985) Processes of birth and death and simple stochastic epidemic models. Autom. Remote Control. 6, 128–135 (in Russian).Google Scholar
[22] Zeifman, A. I. (1985) Stability for continuous-time nonhomogeneous Markov chains. Lecture Notes in Mathematics, 1155, Springer-Verlag, Berlin, 401–414.Google Scholar