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 [Formula: see text] for a class of [Formula: see text] symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic differential equation driven by fractional Brownian motion with Hurst parameter [Formula: see text]. For Wigner-type matrices, we obtain almost sure relative compactness of [Formula: see text] in [Formula: see text] following the approach in [1]; for Wishart-type matrices, we obtain tightness of [Formula: see text] on [Formula: see text] by tightness criterions provided in Appendix B. The limit of [Formula: see text] as [Formula: see text] is also characterized.