Quantitative Universality for the Largest Eigenvalue of Sample Covariance Matrices
Probability
2019-12-12 v1 Statistics Theory
Statistics Theory
Abstract
We prove the first explicit rate of convergence to the Tracy-Widom distribution for the fluctuation of the largest eigenvalue of sample covariance matrices that are not integrable. Our primary focus is matrices of type and the proof follows the Erd\"{o}s-Schlein-Yau dynamical method. We use a recent approach to the analysis of the Dyson Brownian motion from [5] to obtain a quantitative error estimate for the local relaxation flow at the edge. Together with a quantitative version of the Green function comparison theorem, this gives the rate of convergence. Combined with a result of Lee-Schnelli [26], some quantitative estimates also hold for more general separable sample covariance matrices with general diagonal population .
Cite
@article{arxiv.1912.05473,
title = {Quantitative Universality for the Largest Eigenvalue of Sample Covariance Matrices},
author = {Haoyu Wang},
journal= {arXiv preprint arXiv:1912.05473},
year = {2019}
}