RANDOM VECTORS IN THE ISOTROPIC POSITION

MARK RUDELSON

Abstract. Let y be a random vector in ℝ^{n}, satisfying

𝔼 y ⊗ y = id.

Let M be a natural number and let y_{1}, …, y_{M} be independent copies of y. We prove that for some absolute constant C

𝔼 ∥\frac{1}{M} ∑_{i} y_{i} ⊗ y_{i} − id∥ ≤ C · \begin{matrix}√ & \overline{log M} \\ √ & \overline{M} \end{matrix} · (𝔼 ∥y∥^{log M})^{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.