Convergence of non-linear functionals of smoothed empirical processes and kernel density estimates

We consider regularizations by convolution of the empirical process and study the asymptotic behaviour of non-linear functional of this process. Using a result for the same type of non-linear functional of the Brownian bridge, shown in a previous paper, and a strong approximation theorem, we prove several results for the p-deviation in estimation of the derivatives of the density. We also study the asymptotic behaviour of the number of crossings of the smoothed empirical process defined by Yukich and of a modified version of the Kullback deviation.