Rate of convergence of linear functions on the unitary group

J P Keating, F Mezzadri, B Singphu
2010 Journal of Physics A: Mathematical and Theoretical  
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
more » ... ality for linear combinations of eigenvalues of random unitary, matrices, and so appropriate for strongly dependent random variables.
doi:10.1088/1751-8113/44/3/035204 fatcat:q62wwsq4hvf5zi2kdhfcdj7avu