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

In this article, we study high-dimensional behavior of empirical spectral distributions ${L_N(t), tin[0,T]}$ for a class of $Ntimes N$ 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 ${L_N(t), tin[0,T]}_{Ninmathbb N}$ in $C([0,T], mathbf P(mathbb R))$ following the approach in cite{Anderson2010}; for Wishart-type matrices, we obtain tightness of ${L_N(t), tin[0,T]}_{Ninmathbb N}$ on $C([0,T], mathbf P(mathbb R))$ by tightness criterions provided in Appendix ref{subset:tightness argument}. The limit of ${L_N(t), tin[0,T]}$ as $Nto infty$ is also characterised.