STRONG APPROXIMATION FOR SET-INDEXED PARTIAL SUM PROCESSES VIA KMT CONSTRUCTIONS .1.

Let (X(i))i is-an-element-of Z+d be an array of independent identically distributed zero-mean random vectors with values in R(k). When E(\X1\r) < + infinity, for some r > 2, we obtain the strong approximation of the partial sum process (SIGMA(i is-an-element-of nuS)X(i): S is-an-element-of l) by a Gaussian partial sum process (SIGMA(i is-an-element-of nuS)Y(i): S is-an-element l), uniformly over all sets in a certain Vapnik-Cher-vonenkis class l of subsets of [0, 1]d. The most striking result is that both an array (X(i))i is-an-element-of Z+d of i.i.d. random vectors and an array (Y(i))i is-an-element-of Z+d of independent N(0, Var X1)-distributed random vectors may be constructed in such a way that, up to a power of log nu, sup(S is-an-element-of l)]\SIGMA\t is-an-element-of nuS(X(i) - Y(i))\ = O(nu(d-1)/2 OR nu(d/r)) a.s., for any Vapnik-Chervonenkis class l fulfilling the uniform Minkowsky condition. From a 1985 paper of Beck, it is straightforward to prove that such a result cannot be improved, when l is the class of Euclidean balls.