scholarly journals Analysis of the limiting spectral distribution of large dimensional information-plus-noise type matrices

2007 ◽  
Vol 98 (6) ◽  
pp. 1099-1122 ◽  
Author(s):  
R. Brent Dozier ◽  
Jack W. Silverstein
Keyword(s):  
Spectral Distribution ◽  
Limiting Spectral Distribution ◽  
Noise Type

NO EIGENVALUES OUTSIDE THE SUPPORT OF THE LIMITING SPECTRAL DISTRIBUTION OF INFORMATION-PLUS-NOISE TYPE MATRICES

2012 ◽  
Vol 01 (01) ◽  
pp. 1150004 ◽  
Author(s):  
ZHIDONG BAI ◽  
JACK W. SILVERSTEIN
Keyword(s):  
Probability Distribution ◽  
Spectral Distribution ◽  
Correlation Matrix ◽  
Closed Interval ◽  
Empirical Distribution ◽  
Limiting Distribution ◽  
Limiting Spectral Distribution ◽  
Noise Type ◽  
Sample Correlation ◽  
Class Of Matrices

We consider a class of matrices of the form Cn = (1/N)(Rn + σXn)(Rn + σXn)*, where Xn is an n × N matrix consisting of independent standardized complex entries, Rn is an n × N nonrandom matrix, and σ > 0. Among several applications, Cn can be viewed as a sample correlation matrix, where information is contained in [Formula: see text], but each column of Rn is contaminated by noise. As n → ∞, if n/N → c > 0, and the empirical distribution of the eigenvalues of [Formula: see text] converge to a proper probability distribution, then the empirical distribution of the eigenvalues of Cn converges a.s. to a nonrandom limit. In this paper we show that, under certain conditions on Rn, for any closed interval in ℝ+ outside the support of the limiting distribution, then, almost surely, no eigenvalues of Cn will appear in this interval for all n large.

scholarly journals Brown measure and asymptotic freeness of elliptic and related matrices

2019 ◽  
Vol 08 (02) ◽  
pp. 1950007
Author(s):  
Kartick Adhikari ◽  
Arup Bose
Keyword(s):  
Spectral Distribution ◽  
Limiting Spectral Distribution

We show that independent elliptic matrices converge to freely independent elliptic elements. Moreover, the elliptic matrices are asymptotically free with deterministic matrices under appropriate conditions. We compute the Brown measure of the product of elliptic elements. It turns out that this Brown measure is same as the limiting spectral distribution.

scholarly journals On the spectral distribution of large weighted random regular graphs

2014 ◽  
Vol 03 (04) ◽  
pp. 1450015 ◽  
Author(s):  
Leo Goldmakher ◽  
Cap Khoury ◽  
Steven J. Miller ◽  
Kesinee Ninsuwan
Keyword(s):  
Spectral Distribution ◽  
Regular Graphs ◽  
Weighted Graphs ◽  
Spectral Measures ◽  
Limiting Spectral Distribution ◽  
Random Weights ◽  
Large Trees ◽  
The Difference ◽  
The Relationship ◽  
Limiting Spectral Measure

McKay proved the limiting spectral measures of the ensembles of d-regular graphs with N vertices converge to Kesten's measure as N → ∞. Given a large d-regular graph we assign random weights, drawn from some distribution [Formula: see text], to its edges. We study the relationship between [Formula: see text] and the associated limiting spectral distribution obtained by averaging over the weighted graphs. We establish the existence of a unique "eigendistribution" (a weight distribution [Formula: see text] such that the associated limiting spectral distribution is a rescaling of [Formula: see text]). Initial investigations suggested that the eigendistribution was the semi-circle distribution, which by Wigner's Law is the limiting spectral measure for real symmetric matrices. We prove this is not the case, though the deviation between the eigendistribution and the semi-circular density is small (the first seven moments agree, and the difference in each higher moment is O(1/d2)). Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.

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