Some asymptotic results on density estimators by wavelet projections

Let (X-i)(i >= 1) be an i.i.d. sample on R-d having density f. Given a real function phi on R-d with finite variation, and given an integer valued sequence (J(n)), let (f) over cap (n) denote the estimator of f by wavelet projection based on phi and with multiresolution level equal to j(n). We provide exact rates of almost certain convergence to 0 of the quantity sup(x is an element of H) vertical bar(f) over cap (n)(X) - E((f) over cap (n))vertical bar, when n2(-djn)/log n -> infinity and H is a given hypercube of R-d. We then show that, if n2(-djn)/ log n -> c for a constant c > 0, then the quantity sup(x is an element of H)vertical bar(f) over cap (n)(X) -f vertical bar almost surely fails to converge to 0. (C) 2008 Published by Elsevier B.V.