A CLT for a band matrix model

By Greg W. Anderson, Ofer Zeitouni. 2004

225 citations recorded by the source corpus.

A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on systematic combinatorial enumeration, study of generating functions, and concentration inequalities of the Poincaré type. Special cases treated, with an explicit evaluation of limiting variances, are generalized Wigner and Wishart matrices.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4be923aec2d7dc9b5d4c5bcfdc167d84351ce9e9

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile (2004, methodology): “In the Wigner case with a variance profile (matrix Yn and the variance profile are symmetric), let us mention the recent work of Anderson and Zeitouni [1] where the asymptotics of the spectum and central limit theorems are investigated by means of systematic combinatorial enumeration.”
Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane (2015, background): “1) see [24, 37] (in the Gaussian setting), [7] (with the additional condition that the fourth moments have a profile), and [5] (with bounded higher moments).”
Universality of Wigner random matrices: a survey of recent results (2010, background): “We refer the reader to more comprehensive surveys on random matrices, especially the classical book of Mehta [70], the survey of the Riemann-Hilbert approach of Deift [22], the recent book of Anderson, Guionnet and Zeitouni [4] and the forthcoming book of Forrester [53].”
THE EMPIRICAL EIGENVALUE DISTRIBUTION OF A GRAM MATRIX: FROM INDEPENDENCE TO STATIONARITY (2005, methodology): “This case has been studied by Casati and Girko [7], Shlyakhtenko [14,15], Anderson and Zeitouni [1] among others.”
Bulk universality for generalized Wigner matrices (2010, background): “For these ensembles Guionnet [21] and Anderson-Zeitouni [1] proved that the density of the eigenvalues converges to the Wigner semi-circle law.”
Random matrix theory in statistics: A review (2014, result): “Further research proving similar results under a variety of settings has been carried out by Anderson and Zeitouni (2006), Bai and Silverstein (2004), Bai and Yao (2005), Chatterjee (2006), Johansson (1998), Lytova and Pastur (2009) and Shcherbina (2011), among others.”
Free Probability and Random Matrices (2014, methodology): “A very extensive treatment of band matrices (not using the language of free probability, but the quite related Wigner type moment method) was given by Anderson and Zeitouni [7].”
Operator norm consistent estimation of large-dimensional sparse covariance matrices (2008, methodology): “the fluctuation behavior of linear functionals of eigenvalues (see Jonsson (1982), Bai and Silverstein (2004), and Anderson and Zeitouni (2006)). However, until very recently there has been less work in the direction of using these powerful results for the sake of concrete data analysis. Of course, ”
Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices (2012, background): “Zeitouni [1] have developed a central limit theorem for linear statistics of C1(R) test functions for a large class of random matrices with independent, but not necessarily identically distributed entries which includes Wigner and Wishart matrices with entries satisfying a Poincaré inequality.”
Central Limit Theorems for Linear Statistics of Heavy Tailed Random Matrices (2013, background): “…and the assumptions on the entries of the Wigner matrices weakened in [7, 6, 36, 42], a more general model of band matrices was considered in [2] (see also [36, 5] for general covariance matrices), unitary matrices where considered in [31, 23, 45, 22], and Chatterjee developed a general…”
Statistical eigen-inference from large Wishart matrices (2007, methodology): “The recent work of Anderson and Zeitouni (2006), despite its ambiguous title, represents a breakthrough on this and other fronts. Anderson and Zeitouni encode the correction term in the coefficients of a power series that can be directly computed from the limiting moment series of the sample covaria”
Concentration of measure and spectra of random matrices: Applications to correlation matrices, elliptical distributions and beyond (2009, background): “Since this result, there has been a flurry of activity, especially in recent years, concerning the behavior of the largest eigenvalue of sample covariance matrices [21, 50], their fluctuation behavior in the null case [14, 19, 28, 29] and under alternatives [6, 16, 40], as well as fluctuation result”
Central limit theorem for signal-to-interference ratio of reduced rank linear receiver (2008, unknown): “For central limit theorems in other matrix models, we refer to [1].”