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

Abstract

Consider the following local empirical process indexed by $K\in \mathcal{G}$, for fixed h>0 and z∈ℝ^d: $$G_{n}(K,h,z):=\sum_{i=1}^{n}K\Bigl(\frac{Z_{i}-z}{h^{1/d}}\Big)-\mathbb{E}\Bigl(K\Bigl(\frac{Z_{i}-z}{h^{1/d}}\Big)\Big),$$ where the Z_i are i.i.d. on ℝ^d. We provide an extension of a result of Mason (2004). Namely, under mild conditions on $\mathcal{G}$ and on the law of Z₁, 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⊂ℝ^d is a compact set with nonempty interior and where h_n and $\mathfrak{h}_{n}$ satisfy the Csörgő-Révész-Stute conditions.