Rate of convergence of linear functions on the unitary group
Mathematical Physics
2012-07-02 v2 math.MP
Probability
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.
Keywords
Cite
@article{arxiv.1009.0695,
title = {Rate of convergence of linear functions on the unitary group},
author = {J. P. Keating and F. Mezzadri and B. Singphu},
journal= {arXiv preprint arXiv:1009.0695},
year = {2012}
}
Comments
34 pages, 1 figure; corrected typos, added remark 3.3, added 3 references