We consider a class of particle systems that generalizes the eigenvalues of a class of matrix-valued processes, of which the empirical measures converge to deterministic measures as the dimension goes to infinity. In this paper, we obtain central limit theorems (CLTs) to characterize the fluctuations of the empirical measures around the limit measures by using stochastic calculus. As applications, CLTs for Dyson’s Brownian motion and the eigenvalues of Wishart process are recovered under slightly more general initial conditions, and a CLT for the eigenvalues of a symmetric matrix-valued Ornstein-Uhlenbeck process is obtained.
J. Song is supported by Shandong University grant 11140089963041 and National Natural Science Foundation of China grant 12071256. J. Yao is supported by HKSAR-RGC-Grant GRF-17307319.
The authors are grateful to the referees for their valuable and detailed comments.
"High-dimensional central limit theorems for a class of particle systems." Electron. J. Probab. 26 1 - 33, 2021. https://doi.org/10.1214/21-EJP646