Kernel density estimators:: convergence in distribution for weighted sup-norms

Let f(n) denote a kernel density estimator of a bounded continuous density f in the real line. Let Psi(t) be a positive continuous function such that parallel toPsif(beta)parallel to(infinity) < infinity. Under natural smoothness conditions, necessary and sufficient conditions for the sequence root nh(n)/2log(n)(h-1) sup(tis an element ofR)\Psi(t)(f(n)(t) - Ef(n)(t))\(properly centered and normalized) to converge in distribution to the double exponential law are obtained. The proof is based on Gaussian approximation and a (new) limit theorem for weighted sup-norms of a stationary Gaussian process. This extends well known results of Bickel and Rosenblatt to the case of weighted sup-norms, with the sup taken over the whole line. In addition, all other possible limit distributions of the above sequence are identified (subject to some regularity assumptions).