We introduce a new construction of a summation process based on the collection of rectangular subsets of unit d-dimensional cube for a triangular array of independent non-identically distributed variables with d-dimensional index, using the non-uniform grid adapted to the variances of the variables. We investigate its convergence in distribution in some Holder spaces. It turns out that for dimensions greater than 2, the limiting process is not necessarily the standard Brownian sheet. This contrasts with a classical result of Prokhorov for the one-dimensional case.
Vaidotas Zemlys. "A Hölderian FCLT for some multiparameter summation process of independent non-identically distributed random variables." Electron. J. Probab. 13 2259 - 2282, 2008. https://doi.org/10.1214/EJP.v13-590