ACTA issues

On the eigenvalue distribution of some symmetric random matrices

Ferenc Oravecz, Dénes Petz

Acta Sci. Math. (Szeged) 63:3-4(1997), 383-395

Abstract. In 1967 Marchenko and Pastur studied the limit of the eigenvalue distribution of the sum of $p(n)$ rank one random projections in the $n$ dimensional space when $p(n)/n \to a$ as $n\to\infty $. More recently this Marchenko--Pastur distribution occured in the free analogue of the Poisson limit theorem. In this paper we derive a recursive as well as an explicite formula for the moments of the Marchenko--Pastur distribution which turn out to be polynomials of $a$. Moreover, an elementary combinatorial proof is given to the known fact that a variant of the Marchenko--Pastur distribution describes the asymptotical eigenvalue density of sample covariance matrices.

AMS Subject Classification (1991): 15A52, 60F05, 60H25

Received November 18, 1996 and in revised form May 5, 1997. (Registered under 5773/2009.)