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arXiv · 1009.0695

Rate of convergence of linear functions on the unitary group

J. P. Keating·F. Mezzadri·B. Singphu
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Abstract

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 combinations of eigenvalues of random unitary, matrices, and so appropriate for strongly dependent random variables.

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J. P. Keating, F. Mezzadri, B. Singphu. 2010-09-03. Rate of convergence of linear functions on the unitary group. https://doi.org/10.1088/1751-8113/44/3/035204

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