Some asymptotic results on density estimators by wavelet projections

Let (Xi)i≥1 be an i.i.d. sample on Rd having density f. Given a real function ϕ on Rd with finite variation, and given an integer valued sequence (jn), let fˆn denote the estimator of f by wavelet projection based on ϕ and with multiresolution level equal to jn. We provide exact rates of almost certain convergence to 0 of the quantity supx∈H|fˆn(x)−E(fˆn)(x)|, when n2−djn/logn→∞ and H is a given hypercube of Rd. We then show that, if n2−djn/logn→c for a constant c>0, then the quantity supx∈H|fˆn(x)−f| almost surely fails to converge to 0.