On the Rate of Convergence in the Central Limit Theorem for Linear Statistics of Gaussian, Laguerre, and Jacobi Ensembles

TL;DR

This paper establishes an upper bound on the convergence rate to the Gaussian distribution for linear statistics of Gaussian, Laguerre, and Jacobi ensembles using the Riemann-Hilbert approach, enhancing understanding of their probabilistic behavior.

Contribution

It provides a new uniform estimate for characteristic functions of linear statistics in matrix ensembles, leading to explicit convergence rate bounds under the Kolmogorov--Smirnov metric.

Findings

Derived an upper bound on the convergence rate to Gaussian distribution

Provided a uniform estimate for characteristic functions of linear statistics

Applied Riemann-Hilbert techniques to ensemble analysis

Abstract

Under the Kolmogorov--Smirnov metric, an upper bound on the rate of convergence to the Gaussian distribution is obtained for linear statistics of the matrix ensembles in the case of the Gaussian, Laguerre, and Jacobi weights. The main lemma gives an estimate for the characteristic functions of the linear statistics; this estimate is uniform over the growing interval. The proof of the lemma relies on the Riemann--Hilbert approach.