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

Let {Xj;j∈Nd,j≥1} be an i.i.d. random field of square integrable centered random elements in the separable Hilbert space HH and ξn, n∈Nd, be the summation processes based on the collection of sets [0,t1]×⋯×[0,td][0,t1]×⋯×[0,td], 0≤ti≤10≤ti≤1, i=1,…,di=1,…,d. When d≥2d≥2, we characterize the weak convergence of (n1⋯nd)−1/2ξn in the Hölder space Hαo(H) by the finiteness of the weak pp moment of ‖X1‖ for p=(1/2−α)−1p=(1/2−α)−1. This contrasts with the Hölderian FCLT for d=1d=1 and H=RH=R [A. Račkauskas, 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)P(|X1|>t)=o(t−p).