Limit theorems for stationary Markov processes with L 2 -spectral gap
Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 2, p. 396-423

Let (X t ,Y t ) t𝕋 be a discrete or continuous-time Markov process with state space 𝕏× d where 𝕏 is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. (X t ,Y t ) t𝕋 is assumed to be a Markov additive process. In particular, this implies that the first component (X t ) t𝕋 is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process (Y t ) t𝕋 is shown to satisfy the following classical limit theorems: (a) the central limit theorem, (b) the local limit theorem, (c) the one-dimensional Berry-Esseen theorem, (d) the one-dimensional first-order Edgeworth expansion, provided that we have sup t(0,1]𝕋 𝔼 π,0 [|Y t | α ]< with the expected order α with respect to the independent case (up to some ε > 0 for (c) and (d)). For the statements (b) and (d), a Markov nonlattice condition is also assumed as in the independent case. All the results are derived under the assumption that the Markov process (X t ) t𝕋 has an invariant probability distribution π, is stationary and has the 𝕃 2 (π) -spectral gap property (that is, (Xt)t∈ℕ is ρ-mixing in the discrete-time case). The case where (X t ) t𝕋 is non-stationary is briefly discussed. As an application, we derive a Berry-Esseen bound for the M-estimators associated with ρ-mixing Markov chains.

Soit (X t ,Y t ) t𝕋 un processus de Markov en temps discret ou continu et d’espace d’état 𝕏× d 𝕏 est un ensemble mesurable quelconque. Son semi-groupe de transition est supposé additif suivant la seconde composante, i.e. (X t ,Y t ) t𝕋 est un processus additif Markovien. En particulier, ceci implique que la première composante (X t ) t𝕋 est également un processus de Markov. Les marches aléatoires Markoviennes ou les fonctionnelles additives d’un processus de Markov sont des exemples de processus additifs Markoviens. Dans cet article, on montre que le processus (Y t ) t𝕋 satisfait les théorèmes limites classiques suivants : (a) le théorème de la limite centrale, (b) le théorème limite local, (c) le théorème uniforme de Berry-Esseen en dimension un, (d) le développement d’Edgeworth d’ordre un en dimension un, pourvu que la condition de moment sup t(0,1]𝕋 𝔼 π,0 [|Y t | α ]< soit satisfaite, avec l’ordre attendu α du cas indépendant (à un ε > 0 près pour (c) et (d)). Pour les énoncés (b) et (d), il faut ajouter une condition nonlattice comme dans le cas indépendant. Tous les résultats sont obtenus sous l’hypothèse d’un processus de Markov (X t ) t𝕋 admettant une mesure de probabilité invariante π et possédant la propriété de trou spectral sur 𝕃 2 (π) (c’est à dire, (Xt)t∈ℕ est ρ-mélangeante dans le cas du temps discret). Le cas où (X t ) t𝕋 est non-stationnaire est brièvement abordé. Nous appliquons nos résultats pour obtenir une borne de Berry-Esseen pour les M-estimateurs associés aux chaînes de Markov ρ-mélangeantes.

DOI : https://doi.org/10.1214/11-AIHP413
Classification:  60J05,  60F05,  60J25,  60J55,  37A30,  62M05
Keywords: Markov additive process, central limit theorems, Berry-Esseen bound, edgeworth expansion, spectral method, ρ-mixing, M-estimator
@article{AIHPB_2012__48_2_396_0,
     author = {Ferr\'e, D\'eborah and Herv\'e, Lo\"\i c and Ledoux, James},
     title = {Limit theorems for stationary Markov processes with $L^2$-spectral gap},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {2},
     year = {2012},
     pages = {396-423},
     doi = {10.1214/11-AIHP413},
     zbl = {1245.60068},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2012__48_2_396_0}
}
Ferré, Déborah; Hervé, Loïc; Ledoux, James. Limit theorems for stationary Markov processes with $L^2$-spectral gap. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 2, pp. 396-423. doi : 10.1214/11-AIHP413. http://www.numdam.org/item/AIHPB_2012__48_2_396_0/

[1] S. Asmussen. Ruin Probabilities. World Sci. Publishing Co. Inc., River Edge, NJ, 2000. | MR 1794582 | Zbl 1247.91080

[2] S. Asmussen. Applied Probability and Queues, Vol. 51, 2nd edition. Springer-Verlag, New York, 2003. | MR 1978607 | Zbl 1029.60001

[3] S. Asmussen, F. Avram and M. R. Pistorius. Russian and American put options under exponential phase-type Lévy models. Stochastic Process. Appl. 109 (2004) 79-111. | MR 2024845 | Zbl 1075.60037

[4] M. Babillot. Théorie du renouvellement pour des chaînes semi-markoviennes transientes. Ann. Inst. H. Poincaré Probab. Statist. 24 (1988) 507-569. | Numdam | MR 978023 | Zbl 0681.60095

[5] A. Benveniste and J. Jacod. Systèmes de Lévy des processus de Markov. Invent. Math. 21 (1973) 183-198. | MR 343375 | Zbl 0265.60074

[6] J. Bergh and J. Löfström. Interpolation Spaces. An Introduction. Springer-Verlag, Berlin, 1976. | MR 482275 | Zbl 0344.46071

[7] R. N. Bhattacharya. On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Probab. Theory Related Fields 60 (1982) 185-201. | MR 663900 | Zbl 0468.60034

[8] P. Billingsley. Probability and Measure, 3rd edition. John Wiley & Sons Inc., New York, 1995. | Zbl 0649.60001

[9] M. Bladt, B. Meini, M. F. Neuts and B. Sericola. Distributions of reward functions on continuous-time Markov chains. In Matrix-Analytic Methods 39-62. World Sci. Publishing, Adelaide, 2002. | MR 1923878 | Zbl 1015.60064

[10] R. C. Bradley. Basic properties of strong mixing conditions. a survey and some open questions. Probab. Surv. 2 (2005) 107-144. | MR 2178042 | Zbl 1189.60077

[11] R. C. Bradley. Introduction to strong mixing conditions (Volume I). Technical report, Indiana Univ., 2005. | Zbl 1134.60004

[12] L. Breiman. Probability. SIAM, Philadelphia, PA, 1993. | MR 1163370 | Zbl 0753.60001

[13] S. Campanato. Proprietà di una famiglia di spazi funzionali. Ann. Scuola Norm. Sup. Pisa 18 (1964) 137-160. | Numdam | MR 167862 | Zbl 0133.06801

[14] O. Cappé, E. Moulines and T. Rydén. Inference in Hidden Markov Models. Springer, New York, 2005. | MR 2159833 | Zbl 1080.62065

[15] E. Çinlar. Markov additive processes Part II. Probab. Theory Related Fields 24 (1972) 95-121. | Zbl 0236.60048

[16] E. Çinlar. Introduction to Stochastic Processes. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1975. | MR 380912 | Zbl 0341.60019

[17] E. Çinlar. Shock and wear models and Markov additive processes. In The Theory and Applications of Reliability, with Emphasis on Bayesian and Nonparametric Methods, Vol. I 193-214. Academic Press, New York, 1977. | MR 478365 | Zbl 0649.60031

[18] M.-F. Chen. From Markov Chains to Non-equilibrium Particle Systems, 2nd edition. World Sci. Publishing Co. Inc., River Edge, NJ, 2004. | MR 2091955 | Zbl 0753.60055

[19] D. Dehay and J.-F. Yao. On likelihood estimation for discretely observed Markov jump processes. Aust. N. Z. J. Stat. 49 (2007) 93-107. | MR 2345413 | Zbl 1117.62082

[20] J. L. Doob. Stochastic Processes. John Wiley & Sons, New York, 1953. | Zbl 0696.60003

[21] Ī. Ī. Ezhov and A. V. Skorohod. Markov processes which are homogeneous in the second component. I. Theory Probab. Appl. 14 (1969) 1-13. | MR 267640 | Zbl 0281.60067

[22] Ī. Ī. Ezhov and A. V. Skorohod. Markov processes which are homogeneous in the second component. II. Theory Probab. Appl. 14 (1969) 652-667. | MR 267640 | Zbl 0208.44104

[23] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley and Sons, New York, 1971. | MR 270403 | Zbl 0219.60003

[24] D. Ferré. Développement d'Edgeworth d'ordre 1 pour des M-estimateurs dans le cas de chaînes V-géométriquement ergodiques. CRAS 348 (2010) 331-334. | MR 2600134 | Zbl 1186.62103

[25] G. Fort, E. Moulines, G. O. Roberts and J. S. Rosenthal. On the geometric ergodicity of hybrid samplers. J. Appl. Probab. 40 (2003) 123-146. | MR 1953771 | Zbl 1028.65002

[26] C.-D. Fuh and T. L. Lai. Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks. Adv. in Appl. Probab. 33 (2001) 652-673. | MR 1860094 | Zbl 0995.60081

[27] M. Fukushima and M. Hitsuda. On a class of Markov processes taking values on lines and the central limit theorem. Nagoya Math. J. 30 (1967) 47-56. | MR 216568 | Zbl 0178.20603

[28] H. Ganidis, B. Roynette and F. Simonot. Convergence rate of some semi-groups to their invariant probability. Stochastic Process. Appl. 79 (1999) 243-263. | MR 1671843 | Zbl 0962.60073

[29] V. Genon-Catalot, T. Jeantheau and C. Larédo. Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli 6 (2000) 1051-1079. | MR 1809735 | Zbl 0966.62048

[30] P. W. Glynn and W. Whitt. Limit theorems for cumulative processes. Stochastic Process. Appl. 47 (1993) 299-314. | MR 1239842 | Zbl 0779.60021

[31] P. W. Glynn and W. Whitt. Necessary conditions in limit theorems for cumulative processes. Stochastic Process. Appl. 98 (2002) 199-209. | MR 1887533 | Zbl 1059.60025

[32] B. Goldys and B. Maslowski. Exponential ergodicity for stochastic reaction-diffusion equations. In Stochastic Partial Differential Equations and Applications - VII 115-131. Chapman & Hall/CRC, Boca Raton, FL, 2006. | Zbl 1091.35118

[33] B. Goldys and B. Maslowski. Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE's. Ann. Probab. 34 (2006) 1451-1496. | MR 2257652 | Zbl 1121.60066

[34] M. I. Gordin. On the central limit theorem for stationary Markov processes. Soviet Math. Dokl. 19 (1978) 392-394. | Zbl 0395.60057

[35] S. Gouëzel. Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps. Preprint, 2008. | Zbl 1213.37017

[36] S. Gouëzel and C. Liverani. Banach spaces adapted to Anosov systems. Ergodic Theory Dynam. Systems 26 (2006) 189-217. | Zbl 1088.37010

[37] J.-B. Gravereaux and J. Ledoux. Poisson approximation for some point processes in reliability. Adv. in Appl. Probab. 36 (2004) 455-470. | Zbl 1052.60038

[38] S. Grigorescu and G. Opriçan. Limit theorems for J−X processes with a general state space. Probab. Theory Related Fields 35 (1976) 65-73. | Zbl 0336.60062

[39] D. Guibourg and L. Hervé. A renewal theorem for strongly ergodic Markov chains in dimension d≥3 and in the centered case. Potential Anal. 34 (2011) 385-410. | Zbl 1218.60062

[40] Y. Guivarc'H. Application d'un théorème limite local à la transcience et à la récurrence de marches aléatoires. In Théorie du potentiel (Orsay, 1983) 301-332. Lecture Notes in Math. 1096. Springer, Berlin, 1984. | Zbl 0562.60074

[41] Y. Guivarc'H. Limit theorems for random walks and products of random matrices. In Proceedings of the CIMPA-TIFR School on Probability Measures on Groups, Mumbai 2002 257-332. TIFR Studies in Mathematics Series. Tata Institute of Fundamental Research, Mumbai, India, 2002. | Zbl 1247.60009

[42] 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 (1988) 73-98. | Numdam | Zbl 0649.60041

[43] O. Häggström. Acknowledgement of priority concerning “On the central limit theorem for geometrically ergodic Markov chains.” Probab. Theory Related Fields 135 (2006) 470. | Zbl 1090.60508

[44] H. Hennion and L. Hervé. Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Math. 1766. Springer, Berlin, 2001. | Zbl 0983.60005

[45] H. Hennion and L. Hervé. Central limit theorems for iterated random Lipschitz mappings. Ann. Probab. 32 (2004) 1934-1984. | MR 2073182 | Zbl 1062.60017

[46] L. Hervé. Théorème local pour chaînes de Markov de probabilité de transition quasi-compacte. Applications aux chaînes v-géométriquement ergodiques et aux modèles itératifs. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 179-196. | Numdam | MR 2124640 | Zbl 1085.60049

[47] L. Hervé. Vitesse de convergence dans le théorème limite central pour des chaînes de Markov fortement ergodiques. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 280-292. | Numdam | MR 2446324 | Zbl 1178.60051

[48] L. Hervé, J. Ledoux and V. Patilea. A Berry-Esseen theorem on M-estimators for geometrically ergodic Markov chains. Bernoulli (2012). To appear. | MR 2922467 | Zbl 1279.60089

[49] L. Hervé and F. Pène. The Nagaev-Guivarc'h method via the Keller-Liverani theorem. Bull. Soc. Math. France 138 (2010) 415-489. | Numdam | MR 2729019 | Zbl 1205.60133

[50] M. Hitsuda and A. Shimizu. The central limit theorem for additive functionals of Markov processes and the weak convergence to Wiener measure. J. Math. Soc. Japan 22 (1970) 551-566. | MR 273688 | Zbl 0198.22902

[51] H. Holzmann. Martingale approximations for continuous-time and discrete-time stationary Markov processes. Stochastic Process. Appl. 115 (2005) 1518-1529. | MR 2158018 | Zbl 1073.60050

[52] I. A. Ibragimov. A note on the central limit theorem for dependent random variables. Theory Probab. Appl. 20 (1975) 135-141. | MR 362448 | Zbl 0335.60023

[53] I. A. Ibragimov and Y. V. Linnik. Independent and Stationary Sequences of Random Variables. Walters-Noordhoff, the Netherlands, 1971. | MR 322926 | Zbl 0219.60027

[54] M. Jara, T. Komorowski and S. Olla. Limit theorems for additive functionals of a Markov chain. Ann. Appl. Probab. 19 (2009) 2270-2300. | MR 2588245 | Zbl 1232.60018

[55] S. F. Jarner and E. Hansen. Geometric ergodicity of Metropolis algorithms. Stochastic Process. Appl. 85 (2000) 341-361. | MR 1731030 | Zbl 0997.60070

[56] A. Jobert and L. C. G. Rogers. Option pricing with Markov-modulated dynamics. SIAM J. Control Optim. 44 (2006) 2063-2078. | MR 2248175 | Zbl 1158.91380

[57] G. L. Jones. On the Markov chain central limit theorem. Probab. Surv. 1 (2004) 299-320. | MR 2068475 | Zbl 1189.60129

[58] N. V. Kartashov. Determination of the spectral ergodicity exponent for the birth and death process. Ukrain. Math. J. 52 (2000) 1018-1028. | MR 1817318 | Zbl 0973.60091

[59] J. Keilson and D. M. G. Wishart. A central limit theorem for processes defined on a finite Markov chain. Proc. Cambridge Philos. Soc. 60 (1964) 547-567. | MR 169271 | Zbl 0126.33504

[60] G. Keller and C. Liverani. Stability of the spectrum for transfer operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999) 141-152. | Numdam | MR 1679080 | Zbl 0956.37003

[61] C. Kipnis and S. R. S. Varadhan. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 (1986) 1-19. | MR 834478 | Zbl 0588.60058

[62] P. Lezaud. Chernoff and Berry-Esseen inequalities for Markov processes. ESAIM Probab. Stat. 5 (2001) 183-201. | Numdam | MR 1875670 | Zbl 0998.60075

[63] T. M. Liggett. Exponential L2 convergence of attractive reversible nearest particle systems. Ann. Probab. 17 (1989) 403-432. | MR 985371 | Zbl 0679.60093

[64] N. Limnios and G. Opriçan. Semi-Markov Processes and Reliability. Birkhauser Boston Inc., Boston, MA, 2001. | MR 1843923 | Zbl 0990.60004

[65] N. Maigret. Théorème de limite centrale fonctionnel pour une chaî ne de Markov récurrente au sens de Harris et positive. Ann. Inst. H. Poincaré Probab. Statist. 14 (1978) 425-440. | Numdam | MR 523221 | Zbl 0414.60040

[66] M. Maxwell and M. Woodroofe. A local limit theorem for hidden Markov chains. Statist. Probab. Lett. 32 (1997) 125-131. | MR 1436857 | Zbl 0874.60023

[67] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Springer-Verlag, London, 1993. | MR 1287609 | Zbl 0925.60001

[68] S. V. Nagaev. Some limit theorems for stationary Markov chains. Theory Probab. Appl. 11 (1957) 378-406. | Zbl 0078.31804

[69] J. Neveu. Une généralisation des processus à accroissements positifs indépendants. Abh. Math. Sem. Univ. Hambourg 25 (1961) 36-61. | MR 130714 | Zbl 0103.36303

[70] S. Özekici and R. Soyer. Reliability modeling and analysis in random environments. In Mathematical Reliability: An Expository Perspective 249-273. Kluwer Acad. Publ., Boston, MA, 2004. | MR 2065488

[71] A. Pacheco and N. U. Prabhu. Markov-additive processes of arrivals. In Advances in Queueing 167-194. CRC, Boca Raton, FL, 1995. | MR 1395158 | Zbl 0845.60090

[72] A. Pacheco, L. C. Tang and N. U. Prabhu. Markov-Modulated Processes & Semiregenerative Phenomena. World Sci. Publishing, Hackensack, NJ, 2009. | Zbl 1181.60005

[73] M. Peligrad. On the central limit theorem for ρ-mixing sequences of random variables. Ann. Probab. 15 (1987) 1387-1394. | MR 905338 | Zbl 0638.60032

[74] J. Pfanzagl. The Berry-Esseen bound for minimum contrast estimates. Metrika 17 (1971) 81-91. | MR 295467 | Zbl 0216.47805

[75] M. Pinsky. Differential equations with a small parameter and the central limit theorem for functions defined on a finite Markov chain. Probab. Theory Related Fields 9 (1968) 101-111. | MR 228067 | Zbl 0155.24203

[76] B. L. S. P. Rao. On the rate of convergence of estimators for Markov processes. Probab. Theory Related Fields 26 (1973) 141-152. | MR 339420 | Zbl 0248.62039

[77] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer-Verlag, Berlin, 1999. | MR 1725357 | Zbl 0731.60002

[78] G. O. Roberts and J. S. Rosenthal. Geometric ergodicity and hybrid Markov chains. Electron. Commun. Probab. 2 (1997) 13-25. | MR 1448322 | Zbl 0890.60061

[79] G. O. Roberts and J. S. Rosenthal. General state space Markov chains and MCMC algorithms. Probab. Surv. 1 (2004) 20-71. | MR 2095565 | Zbl 1189.60131

[80] G. O. Roberts and R. L. Tweedie. Geometric L2 and L1 convergence are equivalent for reversible Markov chains. J. Appl. Probab. 38A (2001) 37-41. | MR 1915532 | Zbl 1011.60050

[81] M. Rosenblatt. Markov Processes. Structure and Asymptotic Behavior. Springer-Verlag, New York, 1971. | MR 329037 | Zbl 0236.60002

[82] V. T. Stefanov. Exact distributions for reward functions on semi-Markov and Markov additive processes. J. Appl. Probab. 43 (2006) 1053-1065. | MR 2274636 | Zbl 1152.60069

[83] J. L. Steichen. A functional central limit theorem for Markov additive processes with an application to the closed Lu-Kumar network. Stoch. Models 17 (2001) 459-489. | MR 1871234 | Zbl 0997.60030

[84] A. Touati. Théorèmes de limite centrale fonctionnels pour les processus de Markov. Ann. Inst. H. Poincaré Probab. Statist. 19 (1983) 43-55. | Numdam | MR 699977 | Zbl 0511.60029

[85] A. W. Van Der Vaart. Asymptotic Statistics. Cambridge Univ. Press, Cambridge, 1998. | MR 1652247 | Zbl 0910.62001

[86] L. Wu. Essential spectral radius for Markov semigroups. I. Discrete time case. Probab. Theory Related Fields 128 (2004) 255-321. | MR 2031227 | Zbl 1056.60068