Random vectors in the isotropic position

Abstract

Let $y$ be a random vector in \rn, satisfying $$ \Bbb E \, \tens{y} = id. $$ Let $M$ be a natural number and let $y_1 \etc y_M$ be independent copies of $y$. We prove that for some absolute constant $C$ $$ \enor{\frac{1}{M} \sum_i \tens{y_i} - id} \le C \cdot \frac{\sqrt{\log M}}{\sqrt{M}} \cdot \left ( \enor{y}^{\log M} \right )^{1/ \log M}, $$ provided that the last expression is smaller than 1. We apply this estimate to obtain a new proof of a result of Bourgain concerning the number of random points needed to bring a convex body into a nearly isotropic position.