Density estimation for one-dimensional dynamical systems
ESAIM: Probability and Statistics, Tome 5 (2001), pp. 51-76.

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.

Classification : 37D20, 37M10, 37A50, 60G07, 60G10
Mots clés : 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},
     publisher = {EDP-Sciences},
     volume = {5},
     year = {2001},
     mrnumber = {1875664},
     zbl = {1054.60030},
     language = {en},
     url = {http://archive.numdam.org/item/PS_2001__5__51_0/}
}
TY  - JOUR
AU  - Prieur, Clémentine
TI  - Density estimation for one-dimensional dynamical systems
JO  - ESAIM: Probability and Statistics
PY  - 2001
SP  - 51
EP  - 76
VL  - 5
PB  - EDP-Sciences
UR  - http://archive.numdam.org/item/PS_2001__5__51_0/
LA  - en
ID  - PS_2001__5__51_0
ER  - 
%0 Journal Article
%A Prieur, Clémentine
%T Density estimation for one-dimensional dynamical systems
%J ESAIM: Probability and Statistics
%D 2001
%P 51-76
%V 5
%I EDP-Sciences
%U http://archive.numdam.org/item/PS_2001__5__51_0/
%G en
%F PS_2001__5__51_0
Prieur, Clémentine. Density estimation for one-dimensional dynamical systems. ESAIM: Probability and Statistics, Tome 5 (2001), pp. 51-76. http://archive.numdam.org/item/PS_2001__5__51_0/

[1] A. Amroun, Systèmes dynamiques perturbés. Sur une classe de fonctions zéta dynamiques, Thèse de Doctorat de l'Université Paris 6, Spécialité Mathématique (1995).

[2] P. Ango Nze and P. Doukhan, Non-parametric Minimax estimation in a weakly dependent framework I: Quadratic properties. Math. Methods Statist. 5-4 (1996) 404-423. | MR | Zbl

[3] V. Baladi, M. Benedicks and V. Maume-Deschamps, Almost sure rates of mixing for i.i.d. unimodal maps. Ann. E.N.S. (to appear). | EuDML | Numdam | MR | Zbl

[4] A.D. Barbour, R.M. Gerrard and G. Reinert, Iterates of expanding maps. Probab. Theory Related Fields 116 (2000) 151-180. | MR | Zbl

[5] D. Bosq and D. Guégan, Nonparametric estimation of the chaotic function and the invariant measure of a dynamical system. Statist. Probab. Lett. 25 (1995) 201-212. | MR | Zbl

[6] D. Bosq and J.P. Lecoutre, 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] A. Broise, F. Dal'Bo and M. Peigné, Études spectrales d'opérateurs de transfert et applications. Astérisque 238 (1996) Société‰ Math. de France. | Numdam | Zbl

[8] P. Collet, Some ergodic properties of maps of the interval, in dynamical systems, edited by R. Bamon, J.M. Gambaudo and S. Martinez. Hermann, Paris (1996). | MR | Zbl

[9] C. Coulon-Prieur and P. Doukhan, A triangular central limit Theorem under a new weak dependence condition. Statist. Probab. Lett. 47 (2000) 61-68. | MR | Zbl

[10] W. De Melo and S. Van Strien, One-Dimensional Dynamics. Springer-Verlag (1993). | MR | Zbl

[11] P. Doukhan, Mixing: Properties and Examples. Springer Verlag, Lecture Notes in Statist. 85 (1994). | MR | Zbl

[12] P. Doukhan, Models, Inequalities and Limit Theorems for Stationary Sequences, edited by P. Doukhan, G. Oppenheim and M. Taqqu. Birkhaüser (to appear). | MR | Zbl

[13] P. Doukhan and S. Louhichi, A new weak dependence condition and applications to moment inequalities. Stochastic Process. Appl. 84 (1999) 313-342. | MR | Zbl

[14] P. Doukhan and S. Louhichi, Functional estimation of a density under a new weak dependence condition. Scand. J. Statist. 28 (2001) 325-342. | MR | Zbl

[15] A. Lasota and M. Mackey, Probabilistic properties of deterministic systems. Cambridge University Press (1985). | MR | Zbl

[16] A. Lasota and J.A. Yorke, On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973) 481-488. | MR | Zbl

[17] C. Liverani, Decay of correlations for piecewise expanding maps. J. Statist. Phys. 78 (1995) 1111-1129. | MR | Zbl

[18] C. Liverani, Central limit Theorem for deterministic systems, in Proc. of the international Congress on Dynamical Systems, Montevideo 95. Pittman, Res. Notes Math. (1997). | MR | Zbl

[19] J. Maës, Statistique non paramétrique des processus dynamiques réels en temps discret. Thèse de l'Université Paris 6 (1999).

[20] D. Pollard, Convergence of Stochastic Processes. Springer Verlag, Springer Ser. Statist. (1984). | MR | Zbl

[21] R. Prakasa, Nonparametric functional estimation. Academic Press, New York (1983). | MR | Zbl

[22] E. Rio, About the Lindeberg method for strongly mixing sequences. ESAIM: PS 1 (1995) 35-61. www.emath.fr/ps | Numdam | MR | Zbl

[23] E. Rio, Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes. Probab. Theory Related Fields 104 (1996) 255-282. | Zbl

[24] P.M. Robinson, Non parametric estimators for time series. J. Time Ser. Anal. 4-3 (1983) 185-207. | MR | Zbl

[25] M. Rosenblatt, Stochastic curve estimation, in NSF-CBMS Regional Conference Series in Probability and Statistics, Vol. 3 (1991).

[26] W. Rudin, Real and complex analysis. McGraw-Hill Series in Higher Mathematics, Second Edition (1974). | MR | Zbl

[27] M. Viana, Stochastic dynamics of deterministic systems, Instituto de Matematica Pura e Aplicada. IMPA, Vol. 21 (1997).