Soit un processus de Markov en temps discret ou continu et d’espace d’état où est un ensemble mesurable quelconque. Son semi-groupe de transition est supposé additif suivant la seconde composante, i.e. est un processus additif Markovien. En particulier, ceci implique que la première composante 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 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 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 admettant une mesure de probabilité invariante π et possédant la propriété de trou spectral sur (c’est à dire, (Xt)t∈ℕ est ρ-mélangeante dans le cas du temps discret). Le cas où 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.
Let be a discrete or continuous-time Markov process with state space where is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. is assumed to be a Markov additive process. In particular, this implies that the first component 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 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 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 has an invariant probability distribution π, is stationary and has the -spectral gap property (that is, (Xt)t∈ℕ is ρ-mixing in the discrete-time case). The case where 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.
Mots clés : 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}, pages = {396--423}, publisher = {Gauthier-Villars}, volume = {48}, number = {2}, year = {2012}, doi = {10.1214/11-AIHP413}, zbl = {1245.60068}, language = {en}, url = {http://www.numdam.org/articles/10.1214/11-AIHP413/} }
TY - JOUR AU - Ferré, Déborah AU - Hervé, Loïc AU - Ledoux, James TI - Limit theorems for stationary Markov processes with $L^2$-spectral gap JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2012 SP - 396 EP - 423 VL - 48 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/11-AIHP413/ DO - 10.1214/11-AIHP413 LA - en ID - AIHPB_2012__48_2_396_0 ER -
%0 Journal Article %A Ferré, Déborah %A Hervé, Loïc %A Ledoux, James %T Limit theorems for stationary Markov processes with $L^2$-spectral gap %J Annales de l'I.H.P. Probabilités et statistiques %D 2012 %P 396-423 %V 48 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/11-AIHP413/ %R 10.1214/11-AIHP413 %G en %F 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, Tome 48 (2012) no. 2, pp. 396-423. doi : 10.1214/11-AIHP413. http://www.numdam.org/articles/10.1214/11-AIHP413/
[1] Ruin Probabilities. World Sci. Publishing Co. Inc., River Edge, NJ, 2000. | MR | Zbl
.[2] Applied Probability and Queues, Vol. 51, 2nd edition. Springer-Verlag, New York, 2003. | MR | Zbl
.[3] Russian and American put options under exponential phase-type Lévy models. Stochastic Process. Appl. 109 (2004) 79-111. | MR | Zbl
, and .[4] Théorie du renouvellement pour des chaînes semi-markoviennes transientes. Ann. Inst. H. Poincaré Probab. Statist. 24 (1988) 507-569. | Numdam | MR | Zbl
.[5] Systèmes de Lévy des processus de Markov. Invent. Math. 21 (1973) 183-198. | MR | Zbl
and .[6] Interpolation Spaces. An Introduction. Springer-Verlag, Berlin, 1976. | MR | Zbl
and .[7] 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 | Zbl
.[8] Probability and Measure, 3rd edition. John Wiley & Sons Inc., New York, 1995. | Zbl
.[9] Distributions of reward functions on continuous-time Markov chains. In Matrix-Analytic Methods 39-62. World Sci. Publishing, Adelaide, 2002. | MR | Zbl
, , and .[10] Basic properties of strong mixing conditions. a survey and some open questions. Probab. Surv. 2 (2005) 107-144. | MR | Zbl
.[11] Introduction to strong mixing conditions (Volume I). Technical report, Indiana Univ., 2005. | Zbl
.[12] Probability. SIAM, Philadelphia, PA, 1993. | MR | Zbl
.[13] Proprietà di una famiglia di spazi funzionali. Ann. Scuola Norm. Sup. Pisa 18 (1964) 137-160. | Numdam | MR | Zbl
.[14] Inference in Hidden Markov Models. Springer, New York, 2005. | MR | Zbl
, and .[15] Markov additive processes Part II. Probab. Theory Related Fields 24 (1972) 95-121. | Zbl
.[16] Introduction to Stochastic Processes. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1975. | MR | Zbl
.[17] 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 | Zbl
.[18] From Markov Chains to Non-equilibrium Particle Systems, 2nd edition. World Sci. Publishing Co. Inc., River Edge, NJ, 2004. | MR | Zbl
.[19] On likelihood estimation for discretely observed Markov jump processes. Aust. N. Z. J. Stat. 49 (2007) 93-107. | MR | Zbl
and .[20] Stochastic Processes. John Wiley & Sons, New York, 1953. | Zbl
.[21] Markov processes which are homogeneous in the second component. I. Theory Probab. Appl. 14 (1969) 1-13. | MR | Zbl
and .[22] Markov processes which are homogeneous in the second component. II. Theory Probab. Appl. 14 (1969) 652-667. | MR | Zbl
and .[23] An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley and Sons, New York, 1971. | MR | Zbl
.[24] Développement d'Edgeworth d'ordre 1 pour des M-estimateurs dans le cas de chaînes V-géométriquement ergodiques. CRAS 348 (2010) 331-334. | MR | Zbl
.[25] On the geometric ergodicity of hybrid samplers. J. Appl. Probab. 40 (2003) 123-146. | MR | Zbl
, , and .[26] Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks. Adv. in Appl. Probab. 33 (2001) 652-673. | MR | Zbl
and .[27] On a class of Markov processes taking values on lines and the central limit theorem. Nagoya Math. J. 30 (1967) 47-56. | MR | Zbl
and .[28] Convergence rate of some semi-groups to their invariant probability. Stochastic Process. Appl. 79 (1999) 243-263. | MR | Zbl
, and .[29] Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli 6 (2000) 1051-1079. | MR | Zbl
, and .[30] Limit theorems for cumulative processes. Stochastic Process. Appl. 47 (1993) 299-314. | MR | Zbl
and .[31] Necessary conditions in limit theorems for cumulative processes. Stochastic Process. Appl. 98 (2002) 199-209. | MR | Zbl
and .[32] 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
and .[33] Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE's. Ann. Probab. 34 (2006) 1451-1496. | MR | Zbl
and .[34] On the central limit theorem for stationary Markov processes. Soviet Math. Dokl. 19 (1978) 392-394. | Zbl
.[35] Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps. Preprint, 2008. | Zbl
.[36] Banach spaces adapted to Anosov systems. Ergodic Theory Dynam. Systems 26 (2006) 189-217. | Zbl
and .[37] Poisson approximation for some point processes in reliability. Adv. in Appl. Probab. 36 (2004) 455-470. | Zbl
and .[38] Limit theorems for J−X processes with a general state space. Probab. Theory Related Fields 35 (1976) 65-73. | Zbl
and .[39] A renewal theorem for strongly ergodic Markov chains in dimension d≥3 and in the centered case. Potential Anal. 34 (2011) 385-410. | Zbl
and .[40] 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
.[41] 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
.[42] 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
and .[43] Acknowledgement of priority concerning “On the central limit theorem for geometrically ergodic Markov chains.” Probab. Theory Related Fields 135 (2006) 470. | Zbl
.[44] Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Math. 1766. Springer, Berlin, 2001. | Zbl
and .[45] Central limit theorems for iterated random Lipschitz mappings. Ann. Probab. 32 (2004) 1934-1984. | MR | Zbl
and .[46] 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 | Zbl
.[47] 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 | Zbl
.[48] A Berry-Esseen theorem on M-estimators for geometrically ergodic Markov chains. Bernoulli (2012). To appear. | MR | Zbl
, and .[49] The Nagaev-Guivarc'h method via the Keller-Liverani theorem. Bull. Soc. Math. France 138 (2010) 415-489. | Numdam | MR | Zbl
and .[50] 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 | Zbl
and .[51] Martingale approximations for continuous-time and discrete-time stationary Markov processes. Stochastic Process. Appl. 115 (2005) 1518-1529. | MR | Zbl
.[52] A note on the central limit theorem for dependent random variables. Theory Probab. Appl. 20 (1975) 135-141. | MR | Zbl
.[53] Independent and Stationary Sequences of Random Variables. Walters-Noordhoff, the Netherlands, 1971. | MR | Zbl
and .[54] Limit theorems for additive functionals of a Markov chain. Ann. Appl. Probab. 19 (2009) 2270-2300. | MR | Zbl
, and .[55] Geometric ergodicity of Metropolis algorithms. Stochastic Process. Appl. 85 (2000) 341-361. | MR | Zbl
and .[56] Option pricing with Markov-modulated dynamics. SIAM J. Control Optim. 44 (2006) 2063-2078. | MR | Zbl
and .[57] On the Markov chain central limit theorem. Probab. Surv. 1 (2004) 299-320. | MR | Zbl
.[58] Determination of the spectral ergodicity exponent for the birth and death process. Ukrain. Math. J. 52 (2000) 1018-1028. | MR | Zbl
.[59] A central limit theorem for processes defined on a finite Markov chain. Proc. Cambridge Philos. Soc. 60 (1964) 547-567. | MR | Zbl
and .[60] Stability of the spectrum for transfer operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999) 141-152. | Numdam | MR | Zbl
and .[61] Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 (1986) 1-19. | MR | Zbl
and .[62] Chernoff and Berry-Esseen inequalities for Markov processes. ESAIM Probab. Stat. 5 (2001) 183-201. | Numdam | MR | Zbl
.[63] Exponential L2 convergence of attractive reversible nearest particle systems. Ann. Probab. 17 (1989) 403-432. | MR | Zbl
.[64] Semi-Markov Processes and Reliability. Birkhauser Boston Inc., Boston, MA, 2001. | MR | Zbl
and .[65] 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 | Zbl
.[66] A local limit theorem for hidden Markov chains. Statist. Probab. Lett. 32 (1997) 125-131. | MR | Zbl
and .[67] Markov Chains and Stochastic Stability. Springer-Verlag, London, 1993. | MR | Zbl
and .[68] Some limit theorems for stationary Markov chains. Theory Probab. Appl. 11 (1957) 378-406. | Zbl
.[69] Une généralisation des processus à accroissements positifs indépendants. Abh. Math. Sem. Univ. Hambourg 25 (1961) 36-61. | MR | Zbl
.[70] Reliability modeling and analysis in random environments. In Mathematical Reliability: An Expository Perspective 249-273. Kluwer Acad. Publ., Boston, MA, 2004. | MR
and .[71] Markov-additive processes of arrivals. In Advances in Queueing 167-194. CRC, Boca Raton, FL, 1995. | MR | Zbl
and .[72] Markov-Modulated Processes & Semiregenerative Phenomena. World Sci. Publishing, Hackensack, NJ, 2009. | Zbl
, and .[73] On the central limit theorem for ρ-mixing sequences of random variables. Ann. Probab. 15 (1987) 1387-1394. | MR | Zbl
.[74] The Berry-Esseen bound for minimum contrast estimates. Metrika 17 (1971) 81-91. | MR | Zbl
.[75] 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 | Zbl
.[76] On the rate of convergence of estimators for Markov processes. Probab. Theory Related Fields 26 (1973) 141-152. | MR | Zbl
.[77] Continuous Martingales and Brownian Motion, 3rd edition. Springer-Verlag, Berlin, 1999. | MR | Zbl
and .[78] Geometric ergodicity and hybrid Markov chains. Electron. Commun. Probab. 2 (1997) 13-25. | MR | Zbl
and .[79] General state space Markov chains and MCMC algorithms. Probab. Surv. 1 (2004) 20-71. | MR | Zbl
and .[80] Geometric L2 and L1 convergence are equivalent for reversible Markov chains. J. Appl. Probab. 38A (2001) 37-41. | MR | Zbl
and .[81] Markov Processes. Structure and Asymptotic Behavior. Springer-Verlag, New York, 1971. | MR | Zbl
.[82] Exact distributions for reward functions on semi-Markov and Markov additive processes. J. Appl. Probab. 43 (2006) 1053-1065. | MR | Zbl
.[83] 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 | Zbl
.[84] Théorèmes de limite centrale fonctionnels pour les processus de Markov. Ann. Inst. H. Poincaré Probab. Statist. 19 (1983) 43-55. | Numdam | MR | Zbl
.[85] Asymptotic Statistics. Cambridge Univ. Press, Cambridge, 1998. | MR | Zbl
.[86] Essential spectral radius for Markov semigroups. I. Discrete time case. Probab. Theory Related Fields 128 (2004) 255-321. | MR | Zbl
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