In this paper we continue with the study of linear statistics of random Hermitian matrix ensembles. We generalize the result of a previous paper on the probability density function of linear statistics to the finite N ensembles where the interval of support of the eigenvalue spectrum is a single interval. Combining the Hankel determinant formula for orthogonal polynomials which are associated with Hermitian matrix ensembles and the linear statistics theorem for finite N , we obtain the strong

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... oscillatory asymptotics for the polynomials orthogonal with respect to weight functions supported on the real axis. Linear statistics We suppose the probability density function of Q, denoted by P(Q), is known and compute its Fourier transform, P(k) := +∞ −∞ dQe −ikQ P(Q) (2.1) where P(Q) := δ(Q − tr f (M)) M .