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

TL;DR

This work develops a unifying normal-approximation framework by marrying Stein's method with Markov diffusion semigroups through the generator $\mathrm{L}$ and carré du champ $\Gamma$, enabling a multivariate normal limit for functions of a stationary Markov process under a concise set of linear and quadratic conditions. The authors prove Theorem $T:Stein-Markov$, which yields Wasserstein- and $L^1$-type bounds in terms of $\Lambda$, $\Sigma$, and error terms $E_1$, $E_2$, and apply it to the Ornstein–Uhlenbeck process on the Hermitian matrix space to give a short, elementary proof of Johansson's rate for linear eigenvalue statistics. The results unify prior infinitesimal exchangeable-pairs approaches, recover known proofs, and provide a flexible calculus-based method that extends to other ensembles; Chebyshev polynomials naturally arise as the diagonalizing family in the limiting covariance. Overall, the paper delivers a transparent, rate-optimal route to Gaussian fluctuations for linear eigenvalue statistics and clarifies the probabilistic mechanism behind diagonalization via spectral polynomials.

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.

Theorems & Definitions (9)

• Theorem 1: cf. Johansson Johansson2
• Lemma 2
• proof
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• proof : Proof of Theorem \ref{['T:Johansson']}