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.