A nonstandard uniform functional limit law for the increments of the multivariate empirical distribution function

Abstract

Let $(Z_i)_{i\geq 1}$ be an independent, identically distributed sequence of random variables on $\RRR^d$. Under mild conditions on the density of $Z_1$, we provide a nonstandard uniform functional limit law for the following processes on $[0,1)^d$: $$\Delta_n(z,h_n,\cdot):=s\mapsto \frac{\sliin 1_{[0,s_1]\times...\times[0,s_d]}\poo\frac{Z_i-z}{h_n^{1/d}}\pff}{c\log n},\;s\in [0,1)^d,$$ along a sequence $(h_n)\suite$ fulfilling $h_n\downarrow 0,\;nh_n\uparrow,\;nh_n/\log c\rar c>0$. Here $z$ ranges through a compact set of $\RRR^d$. This result is an extension of a theorem of Deheuvels and Mason (1992) to the multivariate, non uniform case.\lb