Uniformité en h dans la loi fonctionnelle limite uniforme les accroissements du processus empirique indéxé par des fonctions

Let (Zi)i⩾1 be an i.i.d. sequence being such that Z1 has a continuous, strictly positive density f on an open subset O⊂Rd. Let H⊂O be a compact subset with nonempty interior and let G be a class of real Borel functions on Rd. For each z∈H and h>0, we set the following G-indexed stochastic process: Gn(K,h,z):=∑i=1nK(Zi−zh1/d)−E(K(Zi−zh1/d)),K∈G. Let (h˜n)n⩾1 and (hn)n⩾1 be two sequences fulfilling the Csörgő-Révész-Stute conditions and satisfying hn