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

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 $H\in (1\mathrm{/}2,1)$. 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 $N\to \mathrm{\infty }$ is also characterised.