Linear functions on the classical matrix groups

Abstract

Let $M$ be a random matrix in the orthogonal group $Ø_n$, distributed according to Haar measure, and let $A$ be a fixed $n\times n$ matrix over $\R$ such that $\tr(AA^t)=n$. Then the total variation distance of the random variable $\tr(AM)$ to standard normal is bounded by $2\sqrt{3}/(n-1)$, and this rate is sharp up to the constant. Analogous results are obtained for $M$ a random unitary matrix and $A$ a fixed $n\times n$ matrix over $\C$. The proofs are applications of a new abstract normal approximation theorem which extends Stein's method of exchangeable pairs to situations in which continuous symmetries are present.

13 pages, reorganized to include new abstract approximation theorem, typographical errors fixed