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

International audienceWe consider regularizations by convolution of the empirical process and study the asymptotic behaviour of nonlinear functionals of this process. Using a result for the same type of non-linear functionals of the Brownian bridge, shown in a previous paper [4], 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 [17] and of a modified version of the Kullback deviatio