Consider the following local empirical process indexed by $K\in \mathcal{ G}$, for fixed $h>0$ and $z\in \mathbb{ R}^d$: $$G_n(K,h,z):=\sum_{ i=1}^n K \Bigl(\frac{ Z_i-z}{ h^{ 1/d}}\Big) - \mathbbE \Bigl(K \Bigl(\frac{ Z_i-z}{ h^{ 1/d}}\Big)\Big),$$ where the $Z_i$ are i.i.d. on $\mathbb{ R}^d$. We provide an extension of a result of Mason (2004). Namely, under mild conditions on $\mathcal{ G}$ and on the law of $Z_1$, we establish a uniform functional limit law for the collections of processes $\bigl\{ G_n(\cdot,h_n,z), z\in H, h\in [h_n,\mathfrak{ h}_n]\big\}$, where $H\subset \mathbb{ R}^d$ is a compact set with nonempty interior and where $h_n$ and $\mathfrak{ h}_n$ satisfy the Cs\"{ o}rg\H{ o}-R\'{ e}v\'{ e}sz-Stute conditions.
Categories: math.ST stat.TH