ON THE LINEAR STATISTICS OF HERMITIAN RANDOM MATRICES

In this paper we continue with the study of linear statistics of rando m Hermitian matrix ensembles. We generalize the result of a previous p aper on the probability density function of linear statistics to the f inite N ensembles where the interval of support of the eigenvalue spec trum is a single interval. Combining the Hankel determinant formula fo r orthogonal polynomials which are associated with Hermitian matrix en sembles and the linear statistics theorem for finite N, we obtain the strong or oscillatory asymptotics for the polynomials orthogonal with respect to weight functions supported on the real axis.