Rate of convergence of linear functions on the unitary group

We study the rate of convergence to a normal random variable of the real and imaginary parts of , where U is an N × N random unitary matrix and AN is a deterministic complex matrix. We show that the rate of convergence is O(N−2 + b), with 0 ≤ b <1, depending only on the asymptotic behaviour of the singular values of AN; for example, if the singular values are non-degenerate, different from zero and O(1) as N → ∞, then b = 0. The proof uses a Berry–Esséen inequality for linear combinations of eigenvalues of random unitary matrices, and so appropriate for strongly dependent random variables.