Random vectors in the isotropic position

Let M be a natural number and let y(1),...,y(M) be independent copies of y. We study the question of approximation of the identity operator by finite sums of the tensors y(i) x y(i) We prove that for some absolute constant C E1/M Sigma(i=1)(M) y(i) x y(i)-id less than or equal to C. root logn/root M. (Ey(logM))(1/logM) provided that the last expression is smaller than 1. We apply this estimate to improve a result of Bourgain concerning the number of random points needed to bring a convex body into a nearly isotropic position. (C) 1999 Academic Press.