Strong approximation of eigenvalues of large dimensional Wishart matrices by roots of generalized Laguerre polynomials

[摘要] 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 lim(n,s-->infinity) = n/s = gamma epsilon[0, infinity). As a by-product we obtain all elemantary proof of the Marcenko-Pastur and the semicircle law without relying on combinatorical arguments. (C) 2002 Elsevier Science (USA).

[发布日期] 2002-10-01

[关键词] random matrix theory;Marcenko-Pastur law;semicircle law;Laguerre polynomials;roots of orthogonal polynomials;strong approximation