Spectral convergence for a general class of random matrices

• Published in: Statistics & probability letters Vol. 81; no. 5; pp. 592 - 602
• Main Authors: Rubio, Francisco, Mestre, Xavier
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.05.2011; Elsevier
• Series: Statistics & Probability Letters
• Subjects: Bioinformatics; Computer Science; Distribution theory; Exact sciences and technology; General topics; Life Sciences; Linear inference, regression; Mathematics; Multivariate statistics; Probability and statistics; Probability theory and stochastic processes; Quantitative Methods; Random matrix theory; Random matrix theory Stieltjes transform Multivariate statistics Sample covariance matrix Separable covariance model; Sample covariance matrix; Sciences and techniques of general use; Separable covariance model; Statistics; Stieltjes transform; Sample covariance matrix; Random matrix theory; Multivariate statistics; Separable covariance model; Stieltjes transform; Random matrix; Statistical moment; Eigenvector; Probability theory; Covariance matrix; Multivariate analysis; Mean estimation; Variance; Convergence; Statistical method; t Norm; Covariance; Signal processing
• ISSN: 0167-7152; 1879-2103
• DOI: 10.1016/j.spl.2011.01.004

Let X be an M × N complex random matrix with i.i.d. entries having mean zero and variance 1 / N and consider the class of matrices of the type B = A + R 1 / 2 XTX H R 1 / 2 , where A , R and T are Hermitian nonnegative definite matrices, such that R and T have bounded spectral norm with T being diagonal, and R 1 / 2 is the nonnegative definite square root of R . Under some assumptions on the moments of the entries of X , it is proved in this paper that, for any matrix Θ with bounded trace norm and for each complex z outside the positive real line, Tr [ Θ ( B − z I M ) − 1 ] − δ M ( z ) → 0 almost surely as M , N → ∞ at the same rate, where δ M ( z ) is deterministic and solely depends on Θ , A , R and T . The previous result can be particularized to the study of the limiting behavior of the Stieltjes transform as well as the eigenvectors of the random matrix model B . The study is motivated by applications in the field of statistical signal processing.