Random vectors in the isotropic position
Mark Rudelson
Let $y$ be a random vector in ${\mathbb R}^n$, satisfying $$ \Bbb E \, y \otimes y = id. $$ Let $M$ be a natural number and let $y_1 , \dots , y_M$ be independent copies of $y$. We prove that for some absolute constant $C$ $$ {\Bbb E \, \left \| {\frac{1}{M} \sum_i^M {y_i} \otimes {y_i} - id} \right \| \le C \cdot \frac{\sqrt{\log M}}{\sqrt{M}} \cdot \left( \Bbb E \, \left \|{y} \right \|^{\log M} \right)^{1/ \log M}}, $$ 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.