Eigenvalue distributions of high-dimensional matrix processes driven by fractional Brownian motion

Abstract

In this article, we study high-dimensional behavior of empirical spectral distributions $\{L_N(t),t\in [0,T]\}$ for a class of $N\times N$ symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H\in (1/2,1)$. For Wigner-type matrices, we obtain almost sure relative compactness of $\{L_N(t),t\in [0,T]{\}}_{N\in \mathbb{N}}$ in $C([0,T],\mathbf{P}(\mathbb{R}))$ following the approach in \cite{Anderson2010}; for Wishart-type matrices, we obtain tightness of $\{L_N(t),t\in [0,T]{\}}_{N\in \mathbb{N}}$ on $C([0,T],\mathbf{P}(\mathbb{R}))$ by tightness criterions provided in Appendix \ref{subset:tightness argument}. The limit of $\{L_N(t),t\in [0,T]\}$ as $N\to \infty$ is also characterised.

28 pages