A Hölderian functional central limit theorem for a multi-indexed summation process

Let be an i.i.d. random field of square integrable centered random elements in the separable Hilbert space and , , be the summation processes based on the collection of sets [0,t1]x...x[0,td], 0<=ti<=1, i=1,...,d. When d>=2, we characterize the weak convergence of in the Hölder space by the finiteness of the weak p moment of for p=(1/2-[alpha])-1. This contrasts with the Hölderian FCLT for d=1 and [A. Rackauskas, Ch. Suquet, Necessary and sufficient condition for the Lamperti invariance principle, Theory Probab. Math. Statist. 68 (2003) 115-124] where the necessary and sufficient condition is P(X1>t)=o(t-p).