LP ADAPTIVE DENSITY-ESTIMATION IN A BETA-MIXING FRAMEWORK

We study the L-pi-integrated risk with pi greater than or equal to 2 o f an adaptive density estimator by wavelets method for absolutely regu lar observations. By a duality argument, the study of the risk is link ed to the control of the supremum of the empirical process over a suit able class of functions. The main argument is a generalization to abso lutely regular variables of a result of Talagrand stated for i.i.d. va riables. Assuming that the sequence of the beta-mixing coefficients (b eta(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.