New Tests of Uniformity on the Compact Classical Groups as Diagnostics for Weak-star Mixing of Markov Chains

TL;DR

This paper develops two new non-parametric goodness-of-fit tests for compact classical groups, providing tools to assess uniformity and mixing times of Markov chains with strong theoretical guarantees.

Contribution

It introduces novel tests for eigenvalue and group uniformity, proves their asymptotic properties, and applies them to analyze Markov chain mixing times.

Findings

Tests are consistent against all fixed alternatives.

Asymptotic distributions under null and alternatives are derived.

Applied to Markov chains, they support rapid mixing claims.

Abstract

This paper introduces two new families of non-parametric tests of goodness-of-fit on the compact classical groups. One of them is a family of tests for the eigenvalue distribution induced by the uniform distribution, which is consistent against all fixed alternatives. The other is a family of tests for the uniform distribution on the entire group, which is again consistent against all fixed alternatives. We find the asymptotic distribution under the null and general alternatives. The tests are proved to be asymptotically admissible. Local power is derived and the global properties of the power function against local alternatives are explored. The new tests are validated on two random walks for which the mixing-time is studied in the literature. The new tests, and several others, are applied to the Markov chain sampler proposed by \cite{jones2011randomized}, providing strong evidence supporting the claim that the sampler mixes quickly.