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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.
|Translated title of the contribution||Rate of convergence of linear functions on the unitary group|
|Number of pages||27|
|Journal||Journal of Physics A: Mathematical and Theoretical|
|Publication status||Published - Jan 2011|