A limited in bandwidth uniformity for the functional limit law of the increments of the empirical process

Consider the following local empirical process indexed by $K\in \mathcal{G}$, for fixed $h>0$ and $z\in {\mathbb{R}}^d$: $$G_n(K,h,z):=\sum _{i=1}^{n}K\left(\frac{Z_i-z}{h^{1\mathrm{/}d}}\right)-\text{mathbbE}\left(K\right(\frac{Z_i-z}{h^{1\mathrm{/}d}}\left)\right),$$ 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 $\{G_n(\cdot ,h_n,z),z\in H,h\in [h_n,{\mathfrak{h}}_n]\}$, 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.