Startseite Stability of functionals of perturbed Markov chains under the condition of uniform minorization
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Stability of functionals of perturbed Markov chains under the condition of uniform minorization

  • Vitaliy Golomoziy EMAIL logo
Veröffentlicht/Copyright: 7. November 2020
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Abstract

In this paper, we investigate the stability of functionals and trajectories of two different, independent, time-inhomogeneous, discrete-time Markov chains on a general state space. We obtain various stability estimates such as an estimate for a difference in expectations of functionals, L2 stability, and a probability of large deviations. The key condition that is used is the minorization condition on the whole space. We consider different limitations on the functional and on the proximity of two chains. We use the coupling method as a primary technique in our proofs.

MSC 2010: 60J45; 60A05; 60K05

Communicated by Vyacheslav L. Girko


References

[1] C. Andrieu, G. Fort and M. Vihola, Quantitative convergence rates for subgeometric Markov chains, J. Appl. Probab. 52 (2015), no. 2, 391–404. 10.1239/jap/1437658605Suche in Google Scholar

[2] M. Benaïm, F. Bouguet and B. Cloez, Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories, Ann. Appl. Probab. 27 (2017), no. 5, 3004–3049. 10.1214/17-AAP1275Suche in Google Scholar

[3] W. Doeblin, Expose de la theorie des chaines simples constantes de Markov a un nomber fini d’estats, Math. l’Union Interbalkan. 2 (1938), 77–105. Suche in Google Scholar

[4] V. V. Golomoziy, Stability of inhomogeneous Markov chains (in Ukrainian), Vysnik Kyivskogo Univ. 4 (2009), 10–15. Suche in Google Scholar

[5] V. V. Golomoziy, A subgeometric estimate for the stability of time-homogeneous Markov chains, Theory Probab. Math. Statist. 81 (2010), 35–50. 10.1090/S0094-9000-2010-00808-8Suche in Google Scholar

[6] V. V. Golomoziy, An inequality for the coupling moment in the case of two inhomogeneous Markov chains, Theory Probab. Math. Statist. 90 (2014), 43–56. 10.1090/tpms/948Suche in Google Scholar

[7] V. V. Golomoziy, On estimation of expectation of simultaneous renewal time of time-inhomogeneous Markov chains using dominating sequence, Mod. Stoch. Theory Appl. 6 (2019), no. 3, 333–343. 10.15559/19-VMSTA138Suche in Google Scholar

[8] V. V. Golomoziy, Stability Estimates for transition probabilities of time-inhomogeneous Markov chains under the uniform minorization condition (in Ukrainian), Theory Probab. Math. Statist. 101 (2019), 78–92. Suche in Google Scholar

[9] V. V. Golomoziy, Stability estimates for transition probabilities of time-inhomogeneous Markov chains under the condition of the minorization on the whole space, Theory Probab. Math. Statist., to appear. Suche in Google Scholar

[10] V. V. Golomoziy and N. V. Kartashov, On coupling moment integrability for time-inhomogeneous Markov chains, Theory Probab. Math. Statist. 89 (2013), 1–12. 10.1090/S0094-9000-2015-00930-3Suche in Google Scholar

[11] V. V. Golomoziy, N. V. Kartashov and Y. Kartashov, Impact of the stress factor on the price of widow’s pensions. Proofs, Theory Probab. Math. Statist. 92 (2016), 17–22. 10.1090/tpms/979Suche in Google Scholar

[12] V. V. Golomoziy and Y. Mishura, Stability estimates for finite-dimensional distributions of time-inhomogeneous Markov chains, MDPI Math. 8 (2020), 10.3390/math8020174. 10.3390/math8020174Suche in Google Scholar

[13] M. V. Kartashov and V. V. Golomoziy, Maximal coupling procedure and stability of discrete Markov chains. I, Theory Probab. Math. Statist. 86 (2012), 81–92. 10.1090/S0094-9000-2013-00891-6Suche in Google Scholar

[14] M. V. Kartashov and V. V. Golomoziy, Maximal coupling procedure and stability of discrete Markov chains. II, Theory Probab. Math. Statist. 87 (2012), 58–70. 10.1090/S0094-9000-2014-00905-9Suche in Google Scholar

[15] M. V. Kartashov and V. V. Golomoziy, Maximal coupling and stability of discrete Markov chains, Theory Probab. Math. Statist. 91 (2015), 17–27. 10.1090/tpms/963Suche in Google Scholar

[16] N. V. Kartashov, Strong Stable Markov Chains, VSP, Utrecht, 1996. 10.1515/9783110917765Suche in Google Scholar

[17] Y. Kartashov, V. V. Golomoziy and N. Kartashov, The impact of stress factors on the price of widow’s pensions, Modern Problems in Insurance Mathematics, EAA Ser., Springer, Cham (2014), 223–237. 10.1007/978-3-319-06653-0_14Suche in Google Scholar

[18] C. Klüppelberg and S. Pergamenchtchikov, Renewal theory for functionals of a Markov chain with compact state space, Ann. Probab. 31 (2003), no. 4, 2270–2300. 10.1214/aop/1068646385Suche in Google Scholar

[19] T. Lindvall, On coupling of continuous-time renewal processes, J. Appl. Probab. 19 (1982), no. 1, 82–89. 10.2307/3213918Suche in Google Scholar

[20] T. Lindvall, Lectures on the Coupling Method, Wiley Ser. Probab. Math. Stat., John Wiley & Sons, New York, 1992. Suche in Google Scholar

[21] P. Ney, A refinement of the coupling method in renewal theory, Stochastic Process. Appl. 11 (1981), no. 1, 11–26. 10.1016/0304-4149(81)90018-1Suche in Google Scholar

[22] D. Revuz, Markov Chains, North-Holland, Amsterdam, 1975. Suche in Google Scholar

[23] H. Thorisson, The coupling of regenerative processes, Adv. in Appl. Probab. 15 (1983), no. 3, 531–561. 10.2307/1426618Suche in Google Scholar

[24] H. Thorisson, Coupling, Stationarity, and Regeneration, Probab. Appl. (N. Y.), Springer, New York, 2000. 10.1007/978-1-4612-1236-2Suche in Google Scholar

[25] P. Tuominen and R. L. Tweedie, Subgeometric rates of convergence of f-ergodic Markov chains, Adv. in Appl. Probab. 26 (1994), no. 3, 775–798. 10.2307/1427820Suche in Google Scholar

Received: 2020-04-14
Accepted: 2020-06-20
Published Online: 2020-11-07
Published in Print: 2020-12-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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