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

Let {Xj;j epsilon N-d.j >= 1) be an i.i.d. random field of square integrable centered random elements in the separable Hilbert space H and, n epsilon N-d, be the summation processes based on the collection of sets [0. t(1)] x... x [0, t(d)], 0 <= ti <= 1, i = 1,..., d. When d >= 2, we characterize the weak convergence of (n(1)...n(d))(-1/2) xi n in the Holder space H-alpha(0) (H) by the finiteness of the weak p moment of parallel to X-1 parallel to for p = (1/2 - alpha)(-1). This contrasts with the Holderian FCLT for d = 1 and H = R [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(vertical bar X-1 vertical bar > t) = o(t(-P)). (c) 2007 Elsevier B.V. All rights reserved.