Multivariate normal approximation for traces of orthogonal and symplectic matrices

TL;DR

This paper establishes a multivariate normal approximation for traces of Haar-distributed orthogonal and symplectic matrices, providing explicit bounds on total variation distance that depend on matrix size and trace order.

Contribution

It introduces a novel approach using Toeplitz+Hankel determinant identities to derive explicit bounds for the normal approximation of trace vectors.

Findings

Total variation distance bounds depend on matrix size and trace order.

Explicit correction terms are provided for the approximation.

Method leverages Toeplitz+Hankel determinant identities.

Abstract

We show that the distance in total variation between $(\mathrm{Tr}\ U, \frac{1}{\sqrt{2}}\mathrm{Tr}\ U^2, \cdots, \frac{1}{\sqrt{m}}\mathrm{Tr}\ U^m)$ and a real Gaussian vector, where $U$ is a Haar distributed orthogonal or symplectic matrix of size $2n$ or $2n+1$, is bounded by $\Gamma(2\frac{n}{m}+1)^{-\frac{1}{2}}$ times a correction. The correction term is explicit and holds for all $n\geq m^4$, for $m$ sufficiently large. For $n\geq m^3$ we obtain the bound $(\frac{n}{m})^{-c\sqrt{\frac{n}{m}}}$ with an explicit constant $c$. Our method of proof is based on an identity of Toeplitz+Hankel determinants due to Basor and Ehrhardt, see \cite{BE}, which is also used to compute the joint moments of the traces.