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. rando m variables having a smooth distribution function F. The perturbed emp irical distribution function (F) over cap(n)$ is obtained by a convolu tion of the classical empirical distribution function F-n and a sequen ce of kernels, In this paper we investigate the almost sure limiting b ehaviour of (F) over cap(n)$ - F-n. These results are applied to obtai n asymptotic results for perturbed empirical processes as well as asym ptotic results for perturbed empirical U-statistics processes and smoo thed bootstrap empirical processes.