Strong limit theorems for the difference of the perturbed empirical distribution function and the classical empirical distribution function

Let {X(n), n greater than or equal to 1} be a sequence of i.i.d. random variables having a smooth distribution function F. The perturbed empirical distribution function (F) over cap(n)$ is obtained by a convolution of the classical empirical distribution function F-n and a sequence of kernels, In this paper we investigate the almost sure limiting behaviour of (F) over cap(n)$ - F-n. These results are applied to obtain asymptotic results for perturbed empirical processes as well as asymptotic results for perturbed empirical U-statistics processes and smoothed bootstrap empirical processes.