A note on limit theorems for perturbed empirical processes

Let Xi, i[greater-or-equal, slanted] 1, be a sequence of i.i.d.k-valued random variables with common distribution P. Let Hnn[greater-or-equal, slanted]1, be a sequence of distribution functions (d.f.) such that , where H0 is the d.f. of the unit mass at zero. The perturbed empirical d.f. is defined by denotes the associated perturbed empirical probability measure. Strong laws of large numbers and weak invariance principles are obtained for the perturbed empirical processes , , where denotes a class of functions on k. The results extend and generalize those of Winter and Yamato and have applications to non-parametric density estimation.