The aim of this paper is to investigate some uniform ergodicity and strong stability estimates for homogeneous markov chains, which may be considered as a refinement of those established by the authors with respect to a given weight norm. As a general rule, the initial parameter values of the most complex systems are approximately known (they are defined on the basis of statistical methods), which results in errors for the calculus of research characteristics for each studied system. For this, the uniform ergodicity and stability inequalities obtained in this paper make it possible to use them in order to estimate numerically the error of definition for the considered characteristics for small perturbations of the system’s parameters. As an example of application, we study the well-known Lindley process, and a comparison with some results obtained by Kartashov is established.
Similar content being viewed by others
References
D. Aïssani and N. V. Kartashov, “Ergodicity and stability of Markov chains with respect to operator topology in the space of transition kernels,” Dokl. Akad. Nauk Uk. SSR, 12, No. 3, 1–4 (1983).
A. A. Borovkov, “Some rate of convergence estimates in stability theorems,” Theory Probab. Appl., 22, No. 4, 689–699 (1977).
I. C. F. Ipsen and C. D. Meyer, “Uniform stability of Markov chains,” SIAM J. Matrix Anal. Appl., 15, No. 4, 1061–1074 (1994).
V. V. Kalashnikov, Qualitative Analysis of Behavior of Complex Systems by Method of Test Functions [in Russian], Nauka, Moscow (1978).
S. Karlin, A First Course in Stochastic Processes, Academic Press, New York (1968).
N. V. Kartashov, “Strongly stable Markov chains,” J. Sov. Math., 34, 1493–1498 (1986).
N. V. Kartashov, Strong Stable Markov Chains, TBIMC Scientific Publishers, VSP, Utrecht (1996).
S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag (1993).
Z. Mouhoubi and D. Aïssani, “On the quantitatives estimates of the uniform ergodicity for Markov chains,” in: Proceeding of the 8th International Vilnius Conference on Probability Theory and Mathematical Statistics, Vilinius, Lithuania (2002), pp. 10–11.
Z. Mouhoubi and D. Aïssani, “On the uniform ergodicity and strong stability estimates of waiting process,” Bull. Int. Stat. Inst., LX, 97–98 (2003).
J. Neveu, Base Mathématiques du Calcul des Probabilités, Masson et Cie, Paris (1964).
D. Revuz, Markov Chains, North-Holland Elsevier, Mathematical Library, Amsterdam (1984).
G. O. Roberts and R. L. Tweedie, “Bounds on regeneration times and convergence rates for Markov chains,” Stoch. Proc. Appl., No. 80, 211–229 (1999).
D. Stoyan, Comparison Methods for Queues and Other Stochastic Models, John Wiley and Sons, New York (1983).
V. M. Zolotarev, “On the continuity of stochastic sequences generated by recurrent processes,” Theory Probab. Appl., XX, No. 4, 819–832 (1975).
Author information
Authors and Affiliations
Corresponding authors
Additional information
* Supported by grant No. 02-07-90147
Proceedings of the XXVI International Seminar on Stability Problems for Stochastic Models, Sovata-Bai, Romania, August 27 – September 2, 2006.
Rights and permissions
About this article
Cite this article
Mouhoubi, Z., Aïssani, D. Uniform Ergodicity and Strong Stability Estimates of Homogeneous Markov Chains*. J Math Sci 200, 452–461 (2014). https://doi.org/10.1007/s10958-014-1928-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-1928-6