Stein's method and the multivariate CLT for traces of powers on the classical compact groups

TL;DR

This paper extends Stein's method to prove a multivariate central limit theorem for the traces of powers of random matrices from classical compact groups, generalizing previous results on convergence rates.

Contribution

It develops a multivariate CLT for traces of powers of Haar-distributed matrices, expanding Fulman's univariate results to the multivariate setting.

Findings

Established multivariate Gaussian convergence for traces of powers.

Extended Stein's method to multivariate setting for matrix traces.

Provided convergence rate estimates for the joint distribution.

Abstract

Let $M_n$ be a random element of the unitary, special orthogonal, or unitary symplectic groups, distributed according to Haar measure. By a classical result of Diaconis and Shahshahani, for large matrix size $n$, the vector $ (\on{Tr}(M_n), \on{Tr}(M_n^2),..., \on{Tr}(M_n^d))$ tends to a vector of independent (real or complex) Gaussian random variables. Recently, Jason Fulman has demonstrated that for a single power $j$ (which may grow with $n$), a speed of convergence result may be obtained via Stein's method of exchangeable pairs. In this note, we extend Fulman's result to the multivariate central limit theorem for the full vector of traces of powers.