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

Let { X j ; j ∈ N d , j ≥ 1 } be an i.i.d. random field of square integrable centered random elements in the separable Hilbert space H and ξ n , n ∈ N d , be the summation processes based on the collection of sets [ 0 , t 1 ] × ⋯ × [ 0 , t d ] , 0 ≤ t i ≤ 1 , i = 1 , … , d . When d ≥ 2 , we characterize the weak convergence of ( n 1 ⋯ n d ) − 1 / 2 ξ n in the Hölder space H α o ( H ) by the finiteness of the weak p moment of ‖ X 1 ‖ for p = ( 1 / 2 − α ) − 1 . This contrasts with the Hölderian FCLT for d = 1 and H = 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 ( | X 1 | > t ) = o ( t − p ) .