Random Vectors in the Isotropic Position

Letybe a random vector in Rn, satisfyingEy⊗y=id.LetMbe a natural number and lety1, …, yMbe independent copies ofy. We study the question of approximation of the identity operator by finite sums of the tensorsyi⊗yi. We prove that for some absolute constantCE 1M ∑i=1M yi⊗yi−id⩽C·log nM·(E ‖y‖log M)1/log M,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.