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High dimensional statistical inference and random matrices
 IN: PROCEEDINGS OF INTERNATIONAL CONGRESS OF MATHEMATICIANS
, 2006
"... Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of interdependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier half of the last century. Subsequently, random matrix theory ..."
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Cited by 49 (1 self)
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Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of interdependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier half of the last century. Subsequently, random matrix theory (RMT) developed, initially within physics, and more recently widely in mathematics. While some of the central objects of study in RMT are identical to those of multivariate statistics, statistical theory was slow to exploit the connection. However, with vast data collection ever more common, data sets now often have as many or more variables than the number of individuals observed. In such contexts, the techniques and results of RMT have much to offer multivariate statistics. The paper reviews some of the progress to date.
Poisson Statistics for the Largest Eigenvalue of Wigner Random Matrices with Heavy Tails, Elect
 Commun. in Probab. 9
, 2004
"... Abstract We study large Wigner random matrices in the case when the marginal distributions of matrix entries have heavy tails. We prove that the largest eigenvalues of such matrices have Poisson statistics. ..."
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Cited by 29 (0 self)
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Abstract We study large Wigner random matrices in the case when the marginal distributions of matrix entries have heavy tails. We prove that the largest eigenvalues of such matrices have Poisson statistics.
A generalization of Wigner’s law
 Comm. Math. Phys
"... We present a generalization of Wigner's semicircle law: we consider a sequence of probability distributions (p1; p2; : : :), with mean value zero and take an N N real symmetric matrix with entries independently chosen from pN and consider analyze the distribution of eigenvalues. If we normali ..."
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We present a generalization of Wigner's semicircle law: we consider a sequence of probability distributions (p1; p2; : : :), with mean value zero and take an N N real symmetric matrix with entries independently chosen from pN and consider analyze the distribution of eigenvalues. If we normalize this distribution by its dispersion we show that as N!1 for certain pN the distribution weakly converges to a universal distribution. The result is a formula for the moments of the universal distribution in terms of the rate of growth of the kth moment of pN (as a function of N), and describe what this means in terms of the support of the distribution. As a corollary, when pN does not depend on N we obtain Wigner's law: if all moments of a distribution are nite, the distribution of eigenvalues is a semicircle. 1
Concentration of the Spectral Measure for Large Random Matrices with Stable Entries
, 2007
"... We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the larges ..."
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We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value.
Random Matrices in 2D, Laplacian Growth and Operator Theory
, 2008
"... Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own whithin applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. Th ..."
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Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own whithin applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this theory has been extended to the case where the spectrum is twodimensional, or even fractal, with dimensions between 1 and 2. In this article, we review these recent developments and indicate some physical problems where the theory can be applied.
Nonconvex Optimization for Linear System with Pregaussian Matrices and Recovery from Multiple Measurements
, 2010
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