The spectral method and ergodic theorems for general Markov chains

S. V. Nagaev

Izvestiya. Mathematics , Tome 79 (2015) no. 2, pp. 311-345.

Résumé

We study the ergodic properties of Markov chains with an arbitrary state space and prove a geometric ergodic theorem. The method of the proof is new: it may be described as an operator method. Our main result is an ergodic theorem for Harris–Markov chains in the case when the return time to some fixed set has finite expectation. Our conditions for the transition function are more general than those used by Athreya–Ney and Nummelin. Unlike them, we impose restrictions not on the original transition function but on the transition function of an embedded Markov chain constructed from the return times to the fixed set mentioned above. The proof uses the spectral theory of linear operators on a Banach space.

Détail

Keywords: embedded Markov chain, uniform ergodicity, resolvent, spectral method, stationary distribution.

Bibliographie

[1] S. V. Nagaev, “Some limit theorems for stationary Markov chains”, Theory Probab. Appl., 2:4 (1957), 378–406 | DOI | MR | Zbl

[2] S. V. Nagaev, “More exact statement of limit theorems for homogeneous Markov chains”, Theory Probab. Appl., 6:1 (1961), 62–81 | DOI | MR | Zbl

[3] É. Le Page, “Théorèmes limites pour les produits de matrices aléatoires”, Probability measures on groups (Oberwolfach, 1981), Lecture Notes in Math., 928, Springer, Berlin–New York, 1982, 258–303 | DOI | MR | Zbl

[4] J. Rousseau-Egele, “Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux”, Ann. Probab., 11:3 (1983), 772–788 | DOI | MR | Zbl

[5] Y. Guivarc'h, “Application d'un théorème limite local à la transcience et à la récurrence de marches de Markov”, Théorie du potentiel (Orsay, 1983), Lecture Notes in Math., 1096, Springer, Berlin, 1984, 301–332 | DOI | MR | Zbl

[6] Y. Guivarc'h, J. Hardy, “Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov”, Ann. Inst. H. Poincaré Probab. Statist., 24:1 (1988), 73–98 | MR | Zbl

[7] X. Milhaud, A. Raugi, “Étude de l'estimateur du maximum de vraisemblance dans le cas d'un processus auto-régressif: convergence, normalité asymptotique, vitesse de convergence”, Ann. Inst. H. Poincaré Probab. Statist., 25:4 (1989), 383–428 | MR | Zbl

[8] H. Hennion, “Dérivabilité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes à coefficients positifs”, Ann. Inst. H. Poincaré Probab. Statist., 27:1 (1991), 27–59 | MR | Zbl

[9] H. Hennion, “Sur un théorème spectral et son application aux noyaux lipchitziens”, Proc. Amer. Math. Soc., 118:2 (1993), 627–634 | DOI | MR | Zbl

[10] H. Hennion, L. Hervé, Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness, Lecture Notes in Math., 1766, Springer-Verlag, Berlin, 2001, viii+145 pp. | DOI | MR | Zbl

[11] H. Hennion, L. Hervé, “Central limit theorems for iterated random Lipschitz mappings”, Ann. Probab., 32:3 (2004), 1934–1984 | DOI | MR | Zbl

[12] F. Pène, “Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard”, Ann. Appl. Probab., 15:4 (2005), 2331–2392 | DOI | MR | Zbl

[13] Y. Guivarc'h, E. Le Page, “On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks”, Ergodic Theory Dynam. Systems, 28:2 (2008), 423–446 | DOI | MR | Zbl

[14] L. Hervé, F. Pène, “The Nagaev–Guivarc'h method via the Keller–Liverani theorem”, Bull. Soc. Math. France, 138:3 (2010), 415–489 | MR | Zbl

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

[16] T. E. Harris, “The existence of stationary measures for certain Markov processes”, Proceedings of the Third Berkeley symposium on mathematical statistics and probability (1954–1955), v. 2, Univ. of California Press, Berkeley–Los Angeles, 1956, 113–124 | MR | Zbl

[17] K. B. Athreya, P. Ney, “A new approach to the limit theory of recurrent Markov chains”, Trans. Amer. Math. Soc., 245 (1978), 493–501 | DOI | MR | Zbl

[18] E. Nummelin, “A splitting technique for Harris recurrent Markov chains”, Z. Wahrsch. Verw. Gebiete, 43:4 (1978), 309–318 | DOI | MR | Zbl

[19] S. V. Nagaev, “On an ergodic theorem for homogeneous Markov chains”, Soviet Math. Dokl., 25 (1982), 281–284 | MR | Zbl

[20] A. A. Borovkov, Ergodicity and stability of stochastic processes, Wiley Ser. Probab. Statist. Probab. Statist., John Wiley Sons, Ltd., Chichester, 1998, xxiv+585 pp. | MR | MR | Zbl | Zbl

[21] S. V. Nagaev, “Ergodic theorems for homogeneous Markov chains”, Soviet Math. Dokl., 39:3 (1989), 483–486 | MR | Zbl

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

[23] S. V. Nagaev, “Some questions on the theory of homogeneous Markov process with discrete time”, Soviet Math. Dokl., 2 (1961), 867–869 | MR | Zbl

[24] S. V. Nagaev, “Ergodicheskie teoremy dlya markovskikh protsessov s diskretnym vremenem”, Sib. matem. zhurn., 6:2 (1965), 413–432 | MR | Zbl

[25] P. Erdös, W. Feller, H. Pollard, “A property of power series with positive coefficients”, Bull. Amer. Math. Soc., 55:2 (1949), 201–204 | DOI | MR | Zbl

[26] B. A. Sevast'yanov, “An ergodic theorem for Markov processes and its application to telephone systems with refusals”, Theory Probab. Appl., 2:1 (1957), 104–112 | DOI | MR | Zbl

[27] S. Orey, “Recurrent Markov chains”, Pacific J. Math., 9:3 (1959), 805–827 | DOI | MR | Zbl

[28] S. Orey, Lecture notes on limit theorems for Markov chain transition probabilities, Van Nostrand Reinhold Mathematical Studies, 34, Van Nostrand Reinhold Co., London–New York–Toronto, Ont., 1971, viii+108 pp. | MR | Zbl

[29] O. A. Butkovsky, A. Yu. Veretennikov, “On asymptotics for Vaserstein coupling of Markov chains”, Stochastic Process. Appl., 123:9 (2013), 3518–3541 | DOI | MR | Zbl

[30] A. Yu. Veretennikov, “On polynomial mixing and convergence rate for stochastic difference and differential equations”, Theory Probab. Appl., 44:2 (2000), 361–374 | DOI | DOI | MR | Zbl