Limit Theorems for Occupation Rates of Local Empirical Processes

Given a continuous probability measure mu on a Borel set H subset of R-d, we prove a limit theorem for occupation rates of the form mu ({z is an element of H, Delta(n)(center dot, h, z) is an element of F}), where the Delta(n)(center dot, h, z) are normalized versions of local empirical processes indexed by a class of functions G. Under standard structural conditions upon G, and under some regularity conditions upon the law of the sample, we show that, almost surely, those occupation rates converge to those of a Gaussian process, uniformly in h is an element of [h(n), h(n)], where h(n) and h(n) are two deterministic bandwidthsequences, upon which mild assumptions are made.