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

Abstract

A 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).