In this paper we prove a Central Limit Theorem for standard kernel estimates of the invariant density of one-dimensional dynamical systems. The two main steps of the proof of this theorem are the following: the study of rate of convergence for the variance of the estimator and a variation on the Lindeberg-Rio method. We also give an extension in the case of weakly dependent sequences in a sense introduced by Doukhan and Louhichi.
Keywords: dynamical systems, decay of correlations, invariant probability, stationary sequences, Lindeberg theorem, central limit theorem, bias, nonparametric estimation, $s$-weakly and $a$-weakly dependent
@article{PS_2001__5__51_0,
author = {Prieur, Cl\'ementine},
title = {Density estimation for one-dimensional dynamical systems},
journal = {ESAIM: Probability and Statistics},
pages = {51--76},
year = {2001},
publisher = {EDP Sciences},
volume = {5},
mrnumber = {1875664},
zbl = {1054.60030},
language = {en},
url = {https://www.numdam.org/item/PS_2001__5__51_0/}
}
Prieur, Clémentine. Density estimation for one-dimensional dynamical systems. ESAIM: Probability and Statistics, Tome 5 (2001), pp. 51-76. https://www.numdam.org/item/PS_2001__5__51_0/
[1] , Systèmes dynamiques perturbés. Sur une classe de fonctions zéta dynamiques, Thèse de Doctorat de l'Université Paris 6, Spécialité Mathématique (1995).
[2] and, Non-parametric Minimax estimation in a weakly dependent framework I: Quadratic properties. Math. Methods Statist. 5-4 (1996) 404-423. | Zbl | MR
[3] , and, Almost sure rates of mixing for i.i.d. unimodal maps. Ann. E.N.S. (to appear). | Zbl | Numdam | MR | EuDML
[4] , and, Iterates of expanding maps. Probab. Theory Related Fields 116 (2000) 151-180. | Zbl | MR
[5] and, Nonparametric estimation of the chaotic function and the invariant measure of a dynamical system. Statist. Probab. Lett. 25 (1995) 201-212. | Zbl | MR
[6] and, Théorie de l'estimation fonctionnelle. Collection “Économie et statistiques avancées”. Série : École Nationale de la Statistique et de l'Administration Économique et Centre d'Études des Programmes Economiques. Economica (1987).
[7] , and , Études spectrales d'opérateurs de transfert et applications. Astérisque 238 (1996) Société Math. de France. | Zbl | Numdam
[8] , Some ergodic properties of maps of the interval, in dynamical systems, edited by R. Bamon, J.M. Gambaudo and S. Martinez. Hermann, Paris (1996). | Zbl | MR
[9] and, A triangular central limit Theorem under a new weak dependence condition. Statist. Probab. Lett. 47 (2000) 61-68. | Zbl | MR
[10] and, One-Dimensional Dynamics. Springer-Verlag (1993). | Zbl | MR
[11] , Mixing: Properties and Examples. Springer Verlag, Lecture Notes in Statist. 85 (1994). | Zbl | MR
[12] , Models, Inequalities and Limit Theorems for Stationary Sequences, edited by P. Doukhan, G. Oppenheim and M. Taqqu. Birkhaüser (to appear). | Zbl | MR
[13] and, A new weak dependence condition and applications to moment inequalities. Stochastic Process. Appl. 84 (1999) 313-342. | Zbl | MR
[14] and, Functional estimation of a density under a new weak dependence condition. Scand. J. Statist. 28 (2001) 325-342. | Zbl | MR
[15] and, Probabilistic properties of deterministic systems. Cambridge University Press (1985). | Zbl | MR
[16] and, On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973) 481-488. | Zbl | MR
[17] , Decay of correlations for piecewise expanding maps. J. Statist. Phys. 78 (1995) 1111-1129. | Zbl | MR
[18] , Central limit Theorem for deterministic systems, in Proc. of the international Congress on Dynamical Systems, Montevideo 95. Pittman, Res. Notes Math. (1997). | Zbl | MR
[19] , Statistique non paramétrique des processus dynamiques réels en temps discret. Thèse de l'Université Paris 6 (1999).
[20] , Convergence of Stochastic Processes. Springer Verlag, Springer Ser. Statist. (1984). | Zbl | MR
[21] , Nonparametric functional estimation. Academic Press, New York (1983). | Zbl | MR
[22] , About the Lindeberg method for strongly mixing sequences. ESAIM: PS 1 (1995) 35-61. www.emath.fr/ps | Zbl | MR | Numdam
[23] , Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes. Probab. Theory Related Fields 104 (1996) 255-282. | Zbl
[24] , Non parametric estimators for time series. J. Time Ser. Anal. 4-3 (1983) 185-207. | Zbl | MR
[25] , Stochastic curve estimation, in NSF-CBMS Regional Conference Series in Probability and Statistics, Vol. 3 (1991).
[26] , Real and complex analysis. McGraw-Hill Series in Higher Mathematics, Second Edition (1974). | Zbl | MR
[27] , Stochastic dynamics of deterministic systems, Instituto de Matematica Pura e Aplicada. IMPA, Vol. 21 (1997).
