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

Abstract

In this paper, 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 $C([0,T],P(ℝ))$ following the approach in [1]; for Wishart-type matrices, we obtain tightness of ${\{L_N(t),t\in [0,T]\}}_{N\in \mathbb{N}}$ on $C([0,T],P(ℝ))$ by tightness criterions provided in Appendix B. The limit of $\{L_N(t),t\in [0,T]\}$ as $N\to \infty$ is also characterized.