Rate of convergence of linear functions on the unitary group

TL;DR

This paper investigates how quickly the real and imaginary parts of the trace of a deterministic matrix multiplied by a random unitary matrix converge to a normal distribution, depending on the matrix's singular values.

Contribution

It establishes a convergence rate of O(N^{-2 + b}) for the trace's parts, depending on the singular value distribution of the deterministic matrix A.

Findings

Convergence rate is O(N^{-2 + b}) for 0 <= b < 1.

Rate depends on the asymptotic behavior of A's singular values.

Non-degenerate singular values lead to b=0, indicating faster convergence.

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.