Stein's method, Markov processes, and linear eigenvalue statistics of random matrices

2025/09/29 by Grzybowski, David, Meckes, Mark
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2509.25451

Abstract

We show how the infinitesimal exchangeable pairs approach to Stein's method combines naturally with the theory of Markov semigroups. We present a multivariate normal approximation theorem for functions of a random variable invariant with respect to a Markov semigroup. This theorem provides a Wasserstein distance bound in terms of quantities related to the infinitesimal generator of the semigroup. As an application, we deduce a rate of convergence for Johansson's celebrated theorem on linear eigenvalue statistics of Gaussian random matrix ensembles.

Citations

• Sharp total variation rates of convergence for fluctuations of linear statistics of β-ensembles
• A short survey of Stein's method
• Stein's method, heat kernel, and linear functions on the orthogonal\n groups
• Stein's method and the multivariate CLT for traces of powers on the classical compact groups
• Stein's method, heat kernel, and traces of powers of elements of compact Lie groups

Related

• Stein's method for max-stable random vectors
• Stein's method for normal approximation in Wasserstein distances with\n application to the multivariate Central Limit Theorem
• A Simple Approach to Global Regime of the Random Matrix Theory
• Malliavin-Stein Method: a Survey of Recent Developments
• Stein's method and the multivariate CLT for traces of powers on the classical compact groups