Strong Approximation of Eigenvalues of Large Dimensional Wishart Matrices by Roots of Generalized Laguerre Polynomials

The purpose of this note is to establish a link between recent results on asymptotics for classical orthogonal polynomials and random matrix theory. Roughly speaking it is demonstrated that the ith eigenvalue of a Wishart matrix W(I_n,s) is close to the ith zero of an appropriately scaled Laguerre polynomial, when . As a by-product we obtain an elemantary proof of the Mar enko–Pastur and the semicircle law without relying on combinatorical arguments.