Non-Hermitian random matrices with a variance profile (I): deterministic equivalents and limiting ESDs

We give an upper bound on the total variation distance between the linear eigenvalue statistic, properly scaled and centered, of a random matrix with a variance profile and the standard Gaussian random variable. The second-order Poincaré inequality-type result introduced in [S. Chatterjee, Fluctuations of eigenvalues and second order poincaré inequalities, Prob. Theory Rel. Fields 143(1) (2009) 1–40.] is used to establish the bound. Using this bound, we prove central limit theorem for linear eigenvalue statistics of random matrices with different kind of variance profiles. We re-establish some existing results on fluctuations of linear eigenvalue statistics of some well-known random matrix ensembles by choosing appropriate variance profiles.

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Some patterned matrices with independent entries

Random Matrices Theory and Application ◽

10.1142/s2010326321500301 ◽

2020 ◽

pp. 2150030

Author(s):

Arup Bose ◽

Koushik Saha ◽

Priyanka Sen

Keyword(s):

Random Matrices ◽

Spectral Distribution ◽

Finite Variance ◽

Weighted Sum ◽

Hankel Matrices ◽

Special Cases ◽

Common Thread ◽

Different Types ◽

Patterned Matrices ◽

Variance Profile

Patterned random matrices such as the reverse circulant, the symmetric circulant, the Toeplitz and the Hankel matrices and their almost sure limiting spectral distribution (LSD), have attracted much attention. Under the assumption that the entries are taken from an i.i.d. sequence with finite variance, the LSD are tied together by a common thread — the [Formula: see text]th moment of the limit equals a weighted sum over different types of pair-partitions of the set [Formula: see text] and are universal. Some results are also known for the sparse case. In this paper, we generalize these results by relaxing significantly the i.i.d. assumption. For our models, the limits are defined via a larger class of partitions and are also not universal. Several existing and new results for patterned matrices, their band and sparse versions, as well as for matrices with continuous and discrete variance profile follow as special cases.

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A CLT FOR INFORMATION-THEORETIC STATISTICS OF NON-CENTERED GRAM RANDOM MATRICES

Random Matrices Theory and Application ◽

10.1142/s2010326311500109 ◽

2012 ◽

Vol 01 (02) ◽

pp. 1150010 ◽

Cited By ~ 15

Author(s):

WALID HACHEM ◽

MALIKA KHAROUF ◽

JAMAL NAJIM ◽

JACK W. SILVERSTEIN

Keyword(s):

Random Matrices ◽

Radio Channel ◽

Random Variable ◽

Multiple Antenna ◽

Information Theoretic ◽

Variable Formula ◽

Main Motivation ◽

Unit Variance ◽

The Moment ◽

Variance Profile

In this article, we study the fluctuations of the random variable: [Formula: see text] where [Formula: see text], as the dimensions of the matrices go to infinity at the same pace. Matrices Xn and An are respectively random and deterministic N × n matrices; matrices Dn and [Formula: see text] are deterministic and diagonal, with respective dimensions N × N and n × n; matrix Xn = (Xij) has centered, independent and identically distributed entries with unit variance, either real or complex. We prove that when centered and properly rescaled, the random variable [Formula: see text] satisfies a Central Limit Theorem and has a Gaussian limit. The variance of [Formula: see text] depends on the moment [Formula: see text] of the variables Xij and also on its fourth cumulant [Formula: see text]. The main motivation comes from the field of wireless communications, where [Formula: see text] represents the mutual information of a multiple antenna radio channel. This article closely follows the companion article "A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile", Ann. Appl. Probab. (2008) by Hachem et al., however the study of the fluctuations associated to non-centered large random matrices raises specific issues, which are addressed here.

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