Research Article

Density Estimation for One-Dimensional Dynamical Systems

Prieur, Clémentine

Université de Cergy-Pontoise, Laboratoire de Mathématiques, bâtiment A4, Site Saint-Martin, 95011 Cergy-Pontoise Cedex, France; prieur@math.u-cergy.fr.

Abstract

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.

(Received January 19 2001)

(Revised March 9 2001)

(Revised June 12 2001)

(Revised July 3 2001)

(Online publication August 15 2002)

Key Words:

• Dynamical systems;
• decay of correlations;
• invariant probability;
• stationary sequences;
• Lindeberg theorem;
• Central Limit Theorem;
• bias;
• nonparametric estimation;
• s-weakly and a-weakly dependent.

Mathematics Subject Classification:

• 37D20;
• 37M10;
• 37A50;
• 60G07;
• 60G10