STRONG APPROXIMATION FOR SET-INDEXED PARTIAL-SUM PROCESSES, VIA KMT CONSTRUCTIONS-II

Let (X(i))i is-an-element-of Z(+)d be an array of zero-mean independent identically distributed random vectors with values in R(k) with finite variance, and let J be a class of Borel subsets of [0, 1]d. If, for the usual metric, J is totally bounded and has a convergent entropy integral, we obtain a strong invariance principle for an appropriately smoothed version of the partial-sum process {SIGMA(i is-an-element-of nu S)X(i): S is-an-element-of J} with an error term depending only on J and on the tail distribution of X1. In particular, when J is the class of subsets of [0, 1]d with alpha-differentiable boundaries introduced by Dudley, we prove that our result is optimal.