New tests of uniformity on the compact classical groups as diagnostics for weak-$^{*}$ mixing of Markov chains

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. The construction of these tests heavily employs facts and techniques from the representation theory of compact groups. In particular, new Cauchy identities are derived and proved for the characters of compact classical groups, in order to accommodate the computation of the test statistic. 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 Jones, Osipov and Rokhlin [Proc. Natl. Acad. Sci. 108 (2011) 15679–15686], providing strong evidence supporting the claim that the sampler mixes quickly.