Linear functions on the classical matrix groups

Let M be a random matrix in the orthogonal group On, distributed according to Haar measure, and let A be a fixed n x n matrix over R such that Tr(AA t ) = n. Then the total variation distance of the random variable Tr(AM) to a standard normal random variable is bounded by 2√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 x 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.

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