Weighted uniform consistency of kernel density estimators

Let fn denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let Ψ(t) be a positive continuous function such that ‖Ψfβ‖∞<∞ for some 0<β<1/2. Under natural smoothness conditions, necessary and sufficient conditions for the sequence √nhₙᵈ2|log hₙᵈ|\|Ψ(t)(fₙ(t)-Efₙ(t))\|∞ to be stochastically bounded and to converge a.s. to a constant are obtained. Also, the case of larger values of β is studied where a similar sequence with a different norming converges a.s. either to 0 or to +∞, depending on convergence or divergence of a certain integral involving the tail probabilities of Ψ(X). The results apply as well to some discontinuous not strictly positive densities.