Some asymptotic results on density estimators by wavelet projections

Let (Xi)i>=1 be an i.i.d. sample on having density f. Given a real function [phi] on with finite variation, and given an integer valued sequence (jn), let denote the estimator of f by wavelet projection based on [phi] and with multiresolution level equal to jn. We provide exact rates of almost certain convergence to 0 of the quantity , when n2-djn/logn-->[infinity] and H is a given hypercube of . We then show that, if n2-djn/logn-->c for a constant c>0, then the quantity almost surely fails to converge to 0.