Theory of Probability and Mathematical Statistics
(Teoriya Imovirnostei ta Matematychna Statystyka)
An estimate of the expectation of the excess of a renewal sequence generated by a time-inhomogeneous Markov chain if a square-integrable majorizing sequence exists
V. V. Golomozyĭ
Download PDF
Abstract: In this paper, we consider conditions which satisfy finitness of theexpectation of the excess of the renewal sequence of the time-inhomogeneous Markov chain. We consider a chain with an arbitrary state space, and some set C. Within this chain we learn a behaviour of a renewal process - a sequence of moments in which chain is returning to C. Main goal of an article to define numerical estimate for the expectati- on for the renewal sequence excess - a time to wait from the moment t until the new renewal.
Keywords: Coupling theory, coupling method, maximal coupling, discrete Markov chains, stability of distributions
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, The Netherlands, 1996, 138 pp.
3. N.V. Kartashov, Exponential asymptotics of the matrix of Markov renewal (in rus.)
4. E. Nummelin, A splitting technique for Harris recurrent chains, Z. Wahrscheinlichkeitstheorie and
Verw. Geb., 43 (1978), 309-318.
5. E. Nummelin, R.L. Tweedie, Geometric ergodicity and R-positivity for general Markov chains, Ann.
Probab., 6 (1978), 404-420.
6. T. Lindvall, On coupling of discrete renewal sequences, Z. Wahrsch. Verw. Gebiete 48(1979), 57–70.
7. I.N. Kovalenko, N.Y. Kuznecov, Construction of the embeded renewal process for multidimentional queing processes and its application for limit theorems. (in rus.)
8. P. Ney, A refinement of the coupling method in renewal theory, Stochastic Processes Appl., 11(1981), 11-26.
9. E. Numemelin, P. Tuominen, Geometric ergodicity of Harris recurrent Markov chains with appli- catoins to renewal theory, Stoch. Proc. Appls., 12 (1982), 187-202.
10. E. Nummelin, General Irreducible Markov Chains and Nonnegative Operators, Cambridge Universi- ty Press, Cambridge, 1984.
11. V.M. Zolotarev, Modern theory of summing independent randov variables (in rus.)
12. S.T. Rachev, Monzhe-Kantrovich problem about masses movement and its application in stochastic (in rus.)
13. T. Lindvall, Lectures on the Coupling Method, John Wiley and Sons, 1991, 249 с.
14. P. Tuominen, R. Tweedie, Subgeometric rates of convergence of f -ergodic Markov chains,. Adv. in
Appl. Probab., 26(1994), 775–798.
15. P. Tuominen, R.L. Tweedie, Subgeometric rates of convergence of f-ergodic Markov Chains,
Advances in Applied Probability, 26 (1994), 775-798.
16. R.L. Tweedie, J. N. Corcoran, Perfect sampling of ergodic Harris chains, Annals of Applied Probabi-
lity, 11 (2001), n.2, 438-451.
17. H. Thorisson, Coupling, Stationarity, and Regeneration, Springer, New York, 2000, 490 с.
18. S.F. Jarner, G.O. Roberts, Polynomial convergence rates of Markov chains, Annals of Applied
Probability, 12 (2001), 224-247.
19. R. Douc, E. Moulines, J.S. Rosenthal, Quantitative bounds for geometric convergence rates of
Markov chains, Annals of Applied Probability, 14 (2004), 1643-1664.
20. R. Douc, E. Mouliness, J.S. Rothenthal, Quantitative bounds on convergence of Time-
inhomogeneous Markov chains, Annals of Applied Probability, 14(2004), n.4, 1643-1665.
21. R. Douc, E. Moulines, P. Soulier, Practical drift conditions for subgeometric rates of convergence,
Annals of Applied Probability, 14 (2004), n 4, 1353-1377.
22. R. Douc, E. Moulines, P. Soulier, Computable convergence rates for subgeometrically ergodic Markov Chains, Bernoulli, 13 (2007), n 3, 831-848.
23. D.J. Daley, Tight bounds for the renewal function of a random walk, Ann.Probab., 8 (1980), No 3, 615-621.
24. R. Douc, G. Fort, A. Guillin, Subgeometric rates of convergence of f-ergodic strong Markov processes, Stochastic Processes and their Applications, 119 (2009), n 3, 897-923.
25. V.V. Golomoziy̆,Stability of time-inhomogeneous Markov chains (in ukr.)
26. V.V. Golomoziy̆,Subgeometri estimate of the stabilityc of time-homogeneous Markov chians (in ukr.)
27. N.V. Kartashov, Bounding, limits and stability of solutions for time-inhomogeneous pertrubation of a renewal equation on the half line. (in ukr.)
28. N.V. Karatshov, V.V. Golomoziy̆,Mean coupling time for independing discrette renewal processes.(in ukr.)
29. N.V. Kartashov, V.V. Golomoziy̆,Maximal coupling and stability of the discrette Markov chain, I.(in ukr.)
30. N.V. Kartashov, V.V. Golomoziy̆,Maximal coupling and stability of the discrette Markov chain, II.(in ukr.)
31. V.V. Golomoziy̆,.МВ .Kartashovов On coupling moment integrability for time-inhomogeneous
Markov chains, Probability theory and mathematical statistics, 89 (2014), 1-12.
32. V.V. Golomoziy̆,Inequalities for the coupling time for two time-inhomogeneous Markov chains. (in ukr.)
33. N.V.Kartashov, V.V. Golomoziy̆,Maximal coupling and stability of discrette time-inhomogeneous Markov chains. (in ukr.)
34. Y. Kartashov, V. Golomoziy, and 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, pp. 223-237.