WEAK-CONVERGENCE OF SMOOTHED EMPIRICAL PROCESSES

Let X(1), X(2),...be a sequence of i.i.d, random variables with common law P. The nth empirical measure is P-n = n(-1) Sigma(i)(n)=1(delta)X(i) and its smoothed version is ($) over tilde P-n = mu n * P-n for a sequence of random signed measures mu(n) converging weakly in probability to zero, We give conditions for the weak convergence of the smoothed empirical processes root n(($) over tilde P-n - P).