Abstract
We consider a Markov chain with general state space and an embedded Markov chain sampled at the times of successive returns to a subsetA0 of the state space.We assume that the latter chain is uniformly ergodic but the originalMarkov chain need not possess this property.We develop amodification of the spectralmethod and utilize it in proving the central limit theorem for theMarkov chain under consideration.
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References
K. B. Athreya and P. Ney, “A new approach to the limit theory of recurrent Markov chains,” Trans. Amer. Math. Soc. 245, 493 (1978).
E. Bolthausen, “The Berry–Esseen theorem for functionals of discrete Markov chains,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 54, 59 (1980).
E. Bolthausen, “The Berry–Esseen theorem for strongly mixing Harris recurrent Markov chains,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 60, 284 (1982).
K. L. Chung, Markov Chains with Stationary Transition Probabilities (Springer-Verlag, Berlin–Heidelberg–New York, 1967).
R. Cogburn, “The central limit theorem for Markov processes,” Proc. 6th Berkeley Sympos. Math. Statist. Probab. Vol. 2, 485 (Univ. California Press, Berkeley, CA, 1972).
W. Doeblin, “Sur deux problèmes deM. Kolmogoroff concernant les chaînes dénombrables,” Bull. Soc. Math. France 66, 210 (1938) [in French].
Y. Guivarc’h, “Application d’un théorème limite local àla transcience et àla recurrence de marches de Markov,” Théorie du Potentiel. Proc. Colloq. J. Deny (Orsay, France, 1983), 301 (Springer-Verlag, Berlin, 1984) [in French].
Y. Guivarc’h and J. Hardy, “Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov,” Ann. Inst. H. Poincaré. Probab. Statist. 24, 73 (1988) [in French].
Y. Guivarc’h and 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, 423 (2008).
T. E. Harris, “The existence of stationary measures for certain Markov processes,” Proc. 3rd Berkeley Sympos. Math. Statist. Probab. Vol. 2, 113 (Univ. California Press, Berkeley, CA, 1956).
H. Hennion, “Dérivabilitédu plus grand exposant caractéristique des produits de matrices aléatoires indépendantes àcoefficients positifs,” Ann. Inst. H. Poincaré. Probab. Statist. 27, 27 (1991) [in French].
H. Hennion, “Sur un théorème spectral et son application aux noyaux lipschitziens,” Proc. Amer. Math. Soc. 118, 627 (1993) [in French].
H. Hennion and L. Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness (Springer, Berlin, 2001).
H. Hennion and L. Hervé, “Central limit theorems for iterated random Lipschitz mappings,” Ann. Probab. 32, 1934 (2004).
L. Hervéand F. Pène, “The Nagaev–Guivarc’h method via the Keller–Liverani theorem,” Bull. Soc. Math. France 138, 415 (2010).
A. N. Kolmogorov, “A local limit theorem for classicalMarkov chains,” Izv. Akad. Nauk SSSR, Ser. Mat. 13, 281 (1949) [in Russian].
E. Le Page, “Théorèmes limites pour les produits de matrices aléatoires,” ProbabilityMeasures on Groups. Proc 6th Conf. (Oberwolfach, 1981), 258 (Springer-Verlag, Berlin, 1982) [in French].
M. Loève, Probability Theory (Dover,Mineola, NY, 2017).
X. Milhaud and A. Raugi, “Étude de l’estimateur du maximum de vraisemblance dans le cas d’un processus autorégressif: convergence, normalitéasymptotique, vitesse de convergence,” Ann. Inst. H. Poincaré. Probab. Statist. 25, 383 (1989) [in French].
S. V. Nagaev, “On several limit theorems for homogeneous Markov chains,” Teor. Veroyatn. Primen. 2, 138 (1957) [in Russian].
S. V. Nagaev, “Some questions on the theory of homogeneous Markov process with discrete time,” Dokl. Akad. Nauk SSSR 139, 34 (1961) [SovietMath., Dokl. 2, 867 (1961)].
S. V. Nagaev, “More exact statement of limit theorems for homogeneous Markov chains,” Teor. Veroyatn. Primen. 6, 67 (1961) [Theory Probab. Appl. 6, 62 (1962)].
S. V. Nagaev, “Ergodic theorems for discrete-time Markov processes,” Sib. Matem. Zh. 6, 413 (1965) [in Russian].
S. V. Nagaev, “The spectralmethod and ergodic theorems for generalMarkov chains,” Izv. Ross. Akad. Nauk, Ser. Mat. 79, 101 (2015) [Izv. Math. 79, 311 (2015)].
E. Nummelin, “A splitting technique for Harris recurrentMarkov chains,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 43, 309 (1978).
S. Orey, “Recurrent Markov chains,” Pacific J. Math. 9, 805 (1959).
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, 2331 (2005).
V. Romanovsky, “Recherches sur les chaînes de Markoff. Premier mémoir,” Acta Math. 66, 147 (1936) [in French].
J. Rousseau-Egele, “Un théorème d’un limite locale pour une classe de transformations dilatantes et monotones par morceaux,” Ann. Probab. 11, 772 (1983) [in French].
T. A. Sarymsakov, Fundamentals of the Theory of Markov Processes (Gostekhizdat, Moscow, 1954) [in Russian].
S. Kh. Sirazhdinov, “More exact statements of limit theorems forMarkov chains,” Dokl. Akad. Nauk SSSR 84, 1143 (1952) [in Russian].
C. Stone, “On local and ratio limit theorems,” Proc. 5th Berkeley Sympos. Math. Statist. Probab. Vol. 2, Pt. 2, 217 (Univ. California Press, Berkeley, CA, 1967).
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Original Russian Text © S.V. Nagaev, 2018, published in Matematicheskie Trudy, 2018, Vol. 21, No. 1, pp. 73–124.
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Nagaev, S.V. The Central Limit Theorem for Markov Chains with General State Space. Sib. Adv. Math. 28, 265–302 (2018). https://doi.org/10.3103/S1055134418040028
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DOI: https://doi.org/10.3103/S1055134418040028