Stein's Method and the Multivariate CLT for Traces of Powers on the Compact Classical Groups

Christian Döbler,Michael Stolz

doi:10.1214/ejp.v16-960

Abstract

Let $M$ 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 of traces of consecutive powers of $M$ 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.

Highlights

Let Mn be a random element of the unitary, special orthogonal, or unitary symplectic groups, distributed according to Haar measure
While Johansson’s approach had its roots in Szegö’s limit theorem for Toeplitz determinants, Stein used the “exchangeable pairs” version of a set of techniques that he had been developing since the early 1970s and that nowadays is referred to as “Stein’s method”
While producing weaker results on the speed than Stein’s and Johansson’s, his theorems apply to the case that the power Mnd(n) grows with n. His techniques seem more likely to be useful in contexts beyond the standard representations of the classical groups

Summary

The main result

D, consider the r-dimensional (complex or real) random vector. In the orthogonal and symplectic cases, let Z := Zd )T denote an r-dimensional real standard normal random vector. Z is defined as standard complex normal, i.e., there are iid real random variables Xd−r+1, . If n ≥ 2d in the unitary and orthogonal cases, and n ≥ 4d + 1 in the symplectic case, the Wasserstein distance between W and ZΣ is d max By d the scaling properties of the Wasserstein metric, the present result implies d fd (M ) , N(0, 1) = O d d n in the orthogonal and symplectic cases and an analogous result in the unitary case. In the onedimensional special case we recover the rate of convergence that was obtained by Fulman, albeit in Wasserstein rather than Kolmogorov distance

Exchangeable pairs and multivariate normal approximation

Power sums and Laplacians

A useful lemma

Proof of the main result in the unitary case

Proof of the main result in the orthogonal case

Proof of the main result in the symplectic case