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A Fourth Moment Inequality for Functionals of Stationary Processes

Part of:Measure-theoretic ergodic theoryLimit theoremsMarkov processesStochastic processesNonparametric inference

Published online by Cambridge University Press:  14 July 2016

Olivier Durieu
Olivier Durieu*
Affiliation:
Université de Rouen
*
∗ Postal address: Laboratoire de Mathématique Raphaël Salem, UMR 6085 CNRS, Université de Rouen, France. Email address: olivier.durieu@etu.univ-rouen.fr
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Abstract

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In this paper, a fourth moment bound for partial sums of functionals of strongly ergodic Markov chains is established. This type of inequality plays an important role in the study of the empirical process invariance principle. This inequality is specially adapted to the technique of Dehling, Durieu, and Volný (2008). The same moment bound can be proved for dynamical systems whose transfer operator has some spectral properties. Examples of applications are given.

Keywords

Stationary processmoment inequalitystrongly ergodic Markov chaindynamical systemempirical distributioninvariance principle

MSC classification

Secondary:60G10: Stationary processes60J10: Markov chains (discrete-time Markov processes on discrete state spaces)60F17: Functional limit theorems; invariance principles28D05: Measure-preserving transformations62G20: Asymptotic properties

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 4 , December 2008, pp. 1086 - 1096
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Baladi, V. (2000). Positive Transfer Operators and Decay of Correlations (Adv. Ser. Nonlinear Dynamics 16). World Scientific, River Edge, NJ.Google Scholar
[2] Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
[3] Broise, A. (1996). Transformations dilatantes de l'intervalle et théorèmes limites. Études spectrales d'opérateurs de transfert et applications. Astérisque 238, 1–109.Google Scholar
[4] Collet, P., Martinez, S. and Schmitt, B. (2004). Asymptotic distribution of tests for expanding maps of the interval. Ergodic Theory Dynam. Systems 24, 707–722.Google Scholar
[5] Dedecker, J. and Prieur, C. (2007). An empirical central limit theorem for dependent sequences. Stoch. Process. Appl. 117, 121–142.CrossRefGoogle Scholar
[6] Dehling, H. and Philipp, W. (2002). Empirical process techniques for dependent data. In Empirical Process Techniques for Dependent Data, Birkhäuser, Boston, MA, pp. 3–113.Google Scholar
[7] Dehling, H., Durieu, O. and Volný, D. (2008). New techniques for empirical processes of dependent data. Preprint. Available at http://arxiv.org/abs/0806.2941.Google Scholar
[8] Donsker, M. D. (1952). Justification and extension of Doob's heuristic approach to the Komogorov–Smirnov theorems. Ann. Math. Statist. 23, 277–281.Google Scholar
[9] Gordin, M. I. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188, 739–741 (in Russian). English translation: Soviet Math. Dokl. 10, 1174-1176.Google Scholar
[10] Gouëzel, S. (2008). An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Preprint. Available at http://arxiv.org/abs/0809.0658v1.Google Scholar
[11] Hennion, H. and Hervé, L. (2001). Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-compactness (Lecture Notes Math. 1766). Springer, Berlin.Google Scholar
[12] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
[13] Parry, W. and Pollicott, M. (1990). Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188, 268.Google Scholar
[14] Pène, F. (2005). Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard. Ann. Appl. Prob. 15, 2331–2392.CrossRefGoogle Scholar