On the empirical distribution of eigenvalues of a class of large dimensional random matrices
type: Item
instance of: scholarly article
title: On the empirical distribution of eigenvalues of a class of large dimensional random matrices
zbMATH Open document ID: 0833.60038
author: Jack W. Silverstein
author: Zhidong Bai
published in: Journal of Multivariate Analysis
publication date: 1996-03-14T00:00:00Z
review text: The eigenvalue distribution of random matrices \[ H_N (x,y) = A_N (x,y) + {1\over N} \sum^n_{j = 1} \tau_j \xi_j (x) \xi_j (y), \qquad x,y = \overline{1,N},\tag{1} \] where Hermitian \(A_N\), real \(\tau\), and complex \(\xi\) are random is studied in the limit \(n,N \to \infty\), \(n/N \to c> 0\). It is proved that the eigenvalue distribution function (EDF) \(\sigma(\lambda; H_N) \equiv \# \{ \lambda^{(N)}_i \leq \lambda\}N^{-1}\), \(\lambda_i^{(N)}\) are eigenvalues of \(H_N\), converges with probability 1 to a nonrandom \(\sigma (\lambda)\), \[ \lim_{n,N \to \infty, n/N \to c} \sigma(\lambda; H_N) = \sigma(\lambda), \tag{2} \] in the case if \(A_N\), \(\tau\), and \(\xi\) are jointly independent, if there exist nonrandom limits \(\sigma_A (\lambda)\) and \(\sigma_\tau (\lambda)\) of \(\sigma (\lambda; A_N)\) and \(\sigma(\lambda; \text{diag} \{\tau_1, \dots, \tau_n\})\), respectively, and if \(\xi_j(x)\) are independent identically distributed random variables. It is shown that the Stieltjes transform \(m(z) =\) \(\int(\lambda - z)^{-1} d\sigma (\lambda)\) is a unique solution of the equation \[ m(z) = \int d\sigma_A (\lambda) \left [\lambda - z + c \int {\mu d\sigma_\tau (\mu)\over 1 + \mu m (z)} \right ]^{-1}.\tag{3} \] This result could be regarded as a generalization of the pioneering work of \textit{V. Marchenko} and \textit{L. Pastur} [Math. USSR, Sb. 1, 457-483 (1967); translation from Math. Sb., n. Ser. 72(114), 507-536 (1967; Zbl 0152.161)], where the convergence \(\sigma(\lambda; H_N)\to\sigma(\lambda)\) in probability was proved and equation (3) was derived for the ensemble of the form (1) with symmetric nonrandom \(A_N\). However, this generalization is not full in the sense that Marchenko and Pastur have considered the case of arbitrary (not necessarily identically) distributed \(\xi_j(x)\) admitting a dependence that vanishes in the limit \(N \to \infty\). In particular, the vectors \(\vec \xi_j\) can be taken as uniformly distributed over the unit …
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 15B52
zbMATH DE Number: 797148
zbMATH Keywords: random matrices
zbMATH Keywords: eigenvalue distribution
zbMATH Keywords: resolvent technique
zbMATH Keywords: large random matrices
zbMATH Keywords: spectral theory of random matrices
reviewed by: A. Khorunzhy
MaRDI profile type: Publication
OpenAlex ID: W2039953809
DOI: 10.1006/JMVA.1995.1051
description: scientific article; zbMATH DE number 797148