Lp adaptive density estimation in a β mixing framework

We study the L-pi-integrated risk with pi greater than or equal to 2 of an adaptive density estimator by wavelets method for absolutely regular observations. By a duality argument, the study of the risk is linked to the control of the supremum of the empirical process over a suitable class of functions. The main argument is a generalization to absolutely regular variables of a result of Talagrand stated for i.i.d. variables. Assuming that the sequence of the beta-mixing coefficients (beta(l))(l greater than or equal to 0) is arithmetically decreasing, we prove that our estimator is adaptive in a class of Besov spaces with unknown smoothness. (C) Elsevier, Paris.