Limit Theorems for Occupation Rates of Local Empirical Processes

Abstract

Given a continuous probability measure μ on a Borel set \(H\subset \mathbb {R}^{d}\), we prove a limit theorem for occupation rates of the form $$\mu\left( \left\{ z\in H,~{\Delta}_{n}(\cdot,h,z)\in F\right\}\right), $$ where the Δn(⋅,h,z) are normalized versions of local empirical processes indexed by a class of functions \(\mathcal {G}\). Under standard structural conditions upon \(\mathcal {G}\), 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 \(h\in [h_{n},\mathfrak {h}_{n}]\), where hn and \(\mathfrak {h}_{n}\) are two deterministic bandwidth sequences, upon which mild assumptions are made.