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Rate of convergence of linear functions on the unitary group

2010
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Journal of Physics A: Mathematical and Theoretical
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We study the rate of convergence to a normal random variable of the real and imaginary parts of Tr(AU), where U is an N x N random unitary matrix and A 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 A; for example, if the singular values are non-degenerate, different from zero and O(1) as N -> infinity, then b=0. The proof uses a Berry-Esse'en inequality for linear

doi:10.1088/1751-8113/44/3/035204
fatcat:q62wwsq4hvf5zi2kdhfcdj7avu