A note on limit theorems for perturbed empirical processes

Abstract

Let Xi, i⩾ 1, be a sequence of i.i.d.Rk-valued random variables with common distribution P. Let Hnn⩾1, be a sequence of distribution functions (d.f.) such that HnH0, where H0 is the d.f. of the unit mass at zero. The perturbed empirical d.f. is defined by F̃n(x):=n−1Σi⩽nx Hn(x−Xi);P̃n denotes the associated perturbed empirical probability measure. Strong laws of large numbers and weak invariance principles are obtained for the perturbed empirical processes (P̃n−P)(f), f ∈F, where F denotes a class of functions on Rk. The results extend and generalize those of Winter and Yamato and have applications to non-parametric density estimation.