The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile

By Walid Hachem, Philippe Loubaton, Jamal Najim. 2004

52 citations recorded by the source corpus.

Consider a N n random matrix Yn = (Y n ij ) where the entries are given by Y n ij = (i=N;j=n) p n X n ij , the X n ij being centered i.i.d. and : (0; 1) 2 ! (0;1) being a continuous function called a variance prole. Consider now a deterministic N n matrix n = n whose o-diagonal entries are zero. Denote by n the non-centered matrix Yn + n and by N^n = min(N;n). Then under the assumption that limn!1 N = c > 0 and 1 N^n N^n X i=1 i N^n ;( n ii) 2 ! n!1 H(dx;d ); where H is a probability measure, it is proven that the empirical distribution of the eigenvalues of n T converges almost surely in distribution to a non-random probability measure. This measure is characterized in terms of its Stieltjes transform, which is ob- tained with the help of an auxiliary system of equations. This kind of results is of interest in the eld of wireless communication. R esum e. Soit Yn = (Y n ij ) une matrice N n dont les entr ees sont donn ees par Y n ij = (i=N;j=n) p n X n, les X n ij etant des variables al eatoires centr ees, i.i.d. et o u : (0; 1) 2 !

Subfields: Computational Theory and Mathematics, Electrical and Electronic Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f6f50f6d384a609c84843fc2f56b420922316833

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Citations in the corpus

Random matrix theory in statistics: A review (2014, background): “In related developments, Hachem et al. (2006) studied the limiting ESD of the sample covariance matrix corresponding to a p n data matrix Yn whose entries are given by Yjk ¼ sðj=p; k=nÞXjk where Xjk are i.i.d. with mean zero and unit variance and s : ½0;1 ½0;1 -ð0;1Þ is a continuous function.”
ON BILINEAR FORMS BASED ON THE RESOLVENT OF LARGE RANDOM MATRICES (2010, background): “The entries of matrices (Xn), (X n ij ; i, j, n) are complex, independent and identically distributed (i.i.d.) with mean 0 and variance 1, and An = (a n ij) is a deterministic N × n matrix whose spectral norm is bounded in n.”
Analysis of the limiting spectral measure of large random matrices of the separable covariance type (2013, background): “Many contributions showed that the spectral measure of ΣnΣn converges to a deterministic probability measure μ and provided a characterization of this limit measure under various assumptions [8, 19, 4, 11], the weakest being found in [22].”
A CLT FOR INFORMATION-THEORETIC STATISTICS OF GRAM RANDOM MATRICES WITH A GIVEN VARIANCE PROFILE (2007, unknown): “Proof. The first item of the lemma follows from Lemma 2.4-(b) together with Theorem 2.3-(3) in [ 16 ].”
Rectangular random matrices, related convolution (2005, background): “We did not even know if this limit existed, even though some computations have been done in some particular cases [12,15,16].”