Geodesic
On improved bounds and conditions for the convergence of~Markov chains
Izvestiya. Mathematics , Tome 86 (2022) no. 1, pp. 92-125.

Voir la notice de l'article provenant de la source Math-Net.Ru

We continue the work of improving the rate of convergence of ergodic homogeneous Markov chains. The setting is more general than in previous papers: we are able to get rid of the assumption about a common dominating measure and consider the case of inhomogeneous Markov chains as well as more general state spaces. We give examples where the new bound for the rate of convergence is the same as (resp. better than) the classical Markov–Dobrushin inequality.
Keywords: ergodicity, generalization of the Markov–Dobrushin condition, rate of convergence.
Mots-clés : Markov chains 
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A. Yu. Veretennikov; M. A. Veretennikova. On improved bounds and conditions for the convergence of~Markov chains. Izvestiya. Mathematics , Tome 86 (2022) no. 1, pp. 92-125. http://geodesic.mathdoc.fr/item/IM2_2022_86_1_a2/

[1] B. V. Gnedenko, Theory of probability, 6th ed., Gordon and Breach Sci. Publ., Newark, NJ, 1997, xxi+497 pp. | Zbl

[2] A. Kolmogoroff, “Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung”, Math. Ann., 104:1 (1931), 415–458 | DOI | MR | Zbl

[3] A. Veretennikov, “Ergodic Markov processes and Poisson equations (lecture notes)”, Modern problems of stochastic analysis and statistics, Selected contributions in honor of V. Konakov's 70th birthday (Moscow, 2016), Springer Proc. Math. Stat., 208, Springer, Cham, 2017, 457–511 | DOI | MR | Zbl

[4] W. Doeblin, “Exposé de la théorie des chaînes simples constantes de Markoff à un nombre fini d'états”, Rev. Math. Union Interbalkan., 2 (1938), 77–105 | Zbl

[5] J. L. Doob, Stochastic processes, John Wiley Sons, Inc., New York; Chapman Hall, Ltd., 1953, viii+654 pp. | MR | MR | Zbl

[6] A. A. Markov, “Rasprostranenie zakona bolshikh chisel na velichiny, zavisyaschie drug ot druga”, Izv. fiz.-matem. o-va pri Kazan. un-te. Ser. 2, 15:4 (1906), 135–156 | Zbl

[7] A. A. Markov, “Rasprostranenie zakona bolshikh chisel na velichiny, zavisyaschie drug ot druga”, Izbrannye trudy po teorii chisel i teorii veroyatnostei, Izd-vo AN SSSR, L., 1951, 339–362 | MR | Zbl

[8] E. Seneta, Non-negative matrices and Markov chains, Springer Ser. Statist., 2nd ed., Springer-Verlag, New York, 1981, xiii+279 pp. | DOI | MR | Zbl

[9] E. Seneta, “Markov and the creation of Markov chains”, MAM 2006: Markov anniversary meeting, Boson Books, Raleigh, NC, 2006, 1–20 https://www.maths.usyd.edu.au/u/eseneta/senetamcfinal.pdf

[10] F. R. Gantmacher, The theory of matrices, v. 1, 2, Reprint of the 1959 ed., Chelsea Publishing Co., New York, 1998, x+374 pp., ix+276 pp. | MR | Zbl

[11] P. Diaconis, D. Stroock, “Geometric bounds for eigenvalues of Markov chains”, Ann. Appl. Probab., 1:1 (1991), 36–61 | DOI | MR | Zbl

[12] O. A. Butkovsky, A. Yu. Veretennikov, “On asymptotics for Vaserstein coupling of Markov chains”, Stochastic Process. Appl., 123:9 (2013) | DOI | MR | Zbl

[13] D. Griffeath, “A maximal coupling for Markov chains”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 31 (1975), 95–106 | DOI | MR | Zbl

[14] T. Lindvall, Lectures on the coupling method, Corr. reprint of the 1992 ed., Dover Publications, Inc., Mineola, NY, 2002, xiv+257 pp. | MR | Zbl

[15] H. Thorisson, Coupling, stationarity, and regeneration, Probab. Appl. (N. Y.), Springer-Verlag, New York, 2000, xiv+517 pp. | MR | Zbl

[16] L. N. Vasershtein, “Markov processes over denumerable products of spaces describing large system of automata”, Problems Inform. Transmission, 5:3 (1969), 47–52 | MR | Zbl

[17] A. Yu. Veretennikov, M. A. Veretennikova, “On convergence rates for homogeneous Markov chains”, Dokl. Math., 101:1 (2020), 12–15 | DOI | DOI

[18] E. B. Dynkin, Markovskie protsessy, Fizmatgiz, M., 1963, 859 pp. ; E. B. Dynkin, Markov processes, т. 1, 2, Grundlehren Math. Wiss., 121, 122, Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin–Göttingen–Heidelberg, 1965, xii+365 pp., viii+274 с. ; v. 1, 2012, 378 pp. | MR | Zbl | DOI | MR | Zbl

[19] R. L. Dobrushin, “Central limit theorem for nonstationary Markov chains. I”, Theory Probab. Appl., 1:1 (1956), 65–80 | DOI | DOI | MR | MR | Zbl

[20] V. V. Kalashnikov, “Metod skleivaniya, ego razvitie i primeneniya”, Obschie neprivodimye tsepi Markova i neotritsatelnye operatory, Mir, M., 1989, 176–190 | MR

[21] E. Nummelin, General irreducible Markov chains and non-negative operators, Cambridge Tracts in Math., 83, Cambridge Univ. Press, Cambridge, 1984, xi+156 pp. | DOI | MR | MR | Zbl

[22] M. A. Krasnosel'skii, E. A. Lifshits, A. V. Sobolev, Positive linear systems. The method of positive operators, Sigma Ser. Appl. Math., 5, Heldermann Verlag, Berlin, 1989, viii+354 pp. | MR | MR | Zbl | Zbl

[23] M. I. Freidlin, A. D. Wentzell, Random perturbations of dynamical systems, Grundlehren Math. Wiss., 260, Springer-Verlag, New York, 1984, viii+326 pp. | DOI | MR | MR | Zbl | Zbl

[24] V. S. Kozyakin, “On the computational aspects of the theory of joint spectral radius”, Dokl. Math., 80:1 (2009), 487–491 | DOI | MR | Zbl

[25] S. Karlin, H. M. Taylor, A first course in stochastic processes, 2nd ed., Academic Press, New York–London, 1975, xvii+557 pp. | MR | Zbl