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Let{Xn,k, k≤ kn n, kn∈N d } be a triangular array of independent non-identically distributed random variables. Using a non uniform grid adapted to the variances of the Xn,k's, we introduce a new construction of a summation processξn of the triangular array based on the collection of sets(0,t1)×¢¢¢× (0,td), 0≤ ti≤ 1, i= 1,...,d. We investigate its convergence in distribution in some Hölder spaces. It turns out that for d≥ 2 the limiting process is not necessarily the standard Brownian sheet. This contrasts with a classical result of Prokhorov for the case d= 1.

A Hölderian FCLT for some multiparameter summation process of independent non-identically distributed random variables