Exponential inequalities and functional estimations for weak dependent data; Applications to dynamical systems

We estimate density and regression functions for weak dependent data. Using an exponential inequality obtained by Dedecker and Prieur (New dependence coefficients. Examples and applications to statistics, Th. Rel. Fields 132 (2005)) and by the author (Exponential inequalities and estimation of conditional probabilities, in Lect. Notes in Stat., Vol. 187 (Springer, 2006)), we control the deviation between the estimator and the function itself. These results are applied to a large class of dynamical systems and lead to estimations of invariant densities and on the mapping itself.