Abstract
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.
Highlights
Convergence of stochastic processes to some Brownian motion or related process is an important topic in probability theory and mathematical statistics
The first functional central limit theorem by Donsker and Prokhorov states the C[0, 1]-weak convergence of n−1/2ξn to the standard Brownian motion W
If we use the same construction for triangular array {Xn,k, k = 1, .., kn, n ∈ }, where for each n ∈ Xn,k are independent but non-identically distributed, polygonal line process will have vertices (k/kn, Sn(k)) with Sn(k) = Xn,1 + · · · + Xn,k
Summary
Convergence of stochastic processes to some Brownian motion or related process is an important topic in probability theory and mathematical statistics. Ξn converges to a standard Brownian motion if the triangular array satisfies the conditions of the central limit theorem. Note that this process coincides with n−1/2ξn in the special case where Xn,k = n−1/2Xk, with i.i.d. Xk’s. The attempt to introduce adaptive construction for general summation processes was made by Bickel and Wichura [1] They put some restrictions on variance of random variables in triangular array. Bickel and Wichura proved that the process ζn converges in the space D([0, 1]2) to a Brownian sheet, if an,i and bn,j are infinitesimally small and random variables {Xn,i j} satisfy Lindeberg condition. In case of special variance structure of the triangular array as in Bickel and Wichura it is shown that the limiting process is a standard Brownian sheet
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