# High-dimensional central limit theorems for eigenvalue distributions of generalized Wishart processes

## TL;DR

This paper establishes central limit theorems for eigenvalue distributions of generalized Wishart processes and related particle systems, revealing their fluctuations around deterministic limits in high dimensions.

## Contribution

It introduces new CLTs for eigenvalue fluctuations of generalized Wishart processes, extending results to Dyson's Brownian motion and Ornstein-Uhlenbeck matrix processes.

## Key findings

- CLTs for eigenvalue fluctuations of generalized Wishart processes
- Extension of CLTs to Dyson's Brownian motion and Ornstein-Uhlenbeck processes
- Eigenvalue empirical measures converge with quantifiable fluctuations

## Abstract

We consider eigenvalues of generalized Wishart processes as well as particle systems, of which the empirical measures converge to deterministic measures as the dimension goes to infinity. In this paper, we obtain central limit theorems to characterize the fluctuations of the empirical measures around the limit measures by using stochastic calculus. As applications, central limit theorems for the Dyson's Brownian motion and the eigenvalues of the Wishart process are recovered under slightly more general initial conditions, and a central limit theorem for the eigenvalues of a symmetric Ornstein-Uhlenbeck matrix process is obtained.