WEAK-CONVERGENCE OF SMOOTHED EMPIRICAL PROCESSES

Let (X(i))i greater-than-or-equal-to 1 be a sequence of i.i.d. random variables with common law P on R(d), d greater-than-or-equal-to 1. The nth empirical measure is P(n) = n-1 SIGMA(i=1)n delta(Xi) and its smoothed version is P(n) = mu(n)*P(n), where (mu(n))n greater-than-or-equal-to 1 is a sequence of probability measures converging weakly to delta-0. We find sufficient conditions implying [GRAPHICS] where F is a class of functions on R(d). The results yield a CLT for perturbed empirical processes, as well as necessary and sufficient conditions implying a CLT for the perturbed empirical distribution process.