Theory of Probability and Mathematical Statistics
Stability estimates for transition probabilities of time-inhomogeneous Markov chains under the condition of the minorization on the whole space
V. Golomoziy
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Abstract: In this paper we derive stability estimates for transition probabilities of two time-inhomo\-ge\-ne\-ous Markov chains with discrette time on the general state space.
Keywords:
Bibliography: re obtained using two conditions: minorization on the whole space condition which is equivalent to the uniformal mixing, and proximity condition for transition
probabilities. Different types of proximity conditions are considered.
Bibliography: 1. W. Doeblin, Expose de la theorie des chaines simples constantes de Markov a un nomber fini d'estats, Mathematique de l'Union Interbalkanique, 2 (1938), 77-105.
2. N.V. Kartashov, Strong Stable Markov Chains, VSP, Utrecht, 1996.
3. P. Ney, A renement of the coupling method in renewal theory, Stochastic Processes Appl., 11 (1981), 11-26.
4. T. Lindvall, Lectures on the Coupling Method, John Wiley and Sons, New York, 1991.
5. T. Lindvall, On coupling for continuous time renewal processes, J. Appl. Probab., 19 (1982), 82-89.
6. H. Thorisson, The coupling of regenerative processes, Adv. Appl. Probab., 15 (1983), 531-561.
7. H. Thorisson, Coupling, Stationarity, and Regeneration, Springer, New York, 2000.
8. R. Douc, E. Moulines, J. S. Rosenthal, Quantitative bounds for geometric convergence rates of Markov chains, Annals of Applied Probability, 14 (2004), 1643-1664.
9. R. Douc, E. Mouliness, P. Solier, Subgeometric ergodicity of Markov chains, Dependence in Probability and Statistics, (2007), 55-64.
10. R. Douc, E. Moulines, P. Soulier, Computable convergence rates for sub-geometric ergodic Markov chains, Bernoulli, 13 (2007), no. 3, 831-848.
11. V. Golomoziy, Stability of inhomogeneous Markov chains, Vysnik Kyivskogo Universitety, 4 (2009), 10-15. (Ukrainian)
12. V. Golomoziy, A subgeometric estimate of the stability for time-homogeneous Markov chains, Theory of probability and mathematical statistics, 81 (2010), 35-50.
13. N. Kartashov, V. Golomoziy, Maximal coupling procedure and stability of discrete Markov chains. I, Theory of probability and mathematical statistics, 86 (2012), 81-92.
14. N. Kartashov, V. Golomoziy, Maximal coupling procedure and stability of discrete Markov chains. II, Theory of probability and mathematical statistics, 87 (2012), 58-70.
15. V. Golomoziy, N. Kartashov On coupling moment integrability for time-inhomogeneous Markov chains, Theory of probability and mathematical statistics, 89 (2014), 1-12.
16. N. Kartashov, V. Golomoziy, Maximal coupling and stability of discrete non-homogeneous Markov chains, Theory of probability and mathematical statistics, 91 (2015), 17-27.
17. V. Golomoziy, N. Kartashov, Y. Kartashov, Impact of the stress factor on the price of widow's pensions. Proofs, Theory of probability and mathematical statistics, 92 (2016), 17-22.
18. V. Kalashnikov, Estimation of duration of transition regime for complex stochastic systems, Trans. Seminar, VNIISI, Moscow (1980), 63-71.
19. D. Grieath, A maximal coupling for Markov chains, Z. Wahrsch. verw. Gebiete, 31 (1975), 95-106.
20. Y. Kartashov, V. Golomoziy, N. Kartashov, The impact of stress factor on the price of widow's pension, Modern Problems in Insurance Mathematics (D. Silverstrov and A. Martin-Lof, eds.), E. A. A. Series, Springer, 2014, 223-237.
21. D. Silvestrov, Synchronized regenerative processes and upper estimates for rate of convergence in ergodic theorems, Rep. Acad. Sci. Ukraine, Series A, 11 (1980), 22-25.
22. D. Silvestrov, Upper estimators in ergodic theorems for regenerative processes, Elektron. Inform. Kybernetik, 16, no. 8/9, (1980), 461-463.
23. D. Silvestrov, Method of a single probability space in ergodic theorems for regenerative processes, 1-3. Math. Operat. Statist., Ser. Optim. Part 1, 14 (1983), no. 2, 285299, Part 2: 16 (1984), no. 4, 216-231, Part 3: 16 (1984), no. 4, 232-244.
24. D. Silvestrov, Coupling for Markov renewal processes and the rate of convergence in ergodic theorems for processes with semi-Markov switchings. Acta Appl. Math., 34 (1994), 109-124.
25. J. W. Pitman, On coupling of Markov chains, Z. Wahrsch. verw. Gebiete, 35 (1979), 315-322.