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A central limit theorem for the determinant of a Wigner matrix
 Adv. Math
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Random matrices: The universality phenomenon for Wigner ensembles
, 2012
"... In this paper, we survey some recent progress on rigorously etablishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment Theorem and its refinements and applications, including the universality of the sine kernel and the Centr ..."
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Cited by 15 (5 self)
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In this paper, we survey some recent progress on rigorously etablishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment Theorem and its refinements and applications, including the universality of the sine kernel and the Central limit theorem of several spectral parameters. We also take the opportunity here to issue some errata for some of our previous papers in this area.
THE CHARACTERISTIC POLYNOMIAL ON COMPACT GROUPS WITH HAAR MEASURE: SOME EQUALITIES IN LAW
, 2007
"... Abstract. This note presents some equalities in law for ZN: = det(Id− G), where G is an element of a subgroup of the set of unitary matrices of size N, endowed with its unique probability Haar measure. Indeed, under some general conditions, ZN can be decomposed as a product of independent random var ..."
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Cited by 3 (2 self)
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Abstract. This note presents some equalities in law for ZN: = det(Id− G), where G is an element of a subgroup of the set of unitary matrices of size N, endowed with its unique probability Haar measure. Indeed, under some general conditions, ZN can be decomposed as a product of independent random variables, whose laws are explicitly known. Our results can be obtained in two ways: either by a recursive decomposition of the Haar measure (Section 1) or by previous results by Killip and Nenciu ([3]) on orthogonal polynomials with respect to some measure on the unit circle (Section 2). This latter method leads naturally to a study of determinants of a class of principal submatrices. Résumé. Cette note présente quelques égalités en loi pour ZN:= det(Id−G), où G est un sousgroupe de l’ensemble des matrices unitaires de taille N, muni de son unique mesure de Haar normalisée. En effet, sous des conditions assez générales, ZN peut être décomposé comme le produit de variables aléatoires indépendantes, dont on connait la loi explicitement. Notre résultat peut être obtenu de deux manières: soit par une décomposition récursive de la mesure de Haar (Partie 1) soit en utilisant un résultat de Killip et Nenciu ([3]) à propos des polynômes orthogonaux relativement à une certaine mesure sur le cercle unité (Partie 2). Cette dernière méthode nous conduit naturellement à l’étude des déterminants de certaines sousmatrices. In this note, 〈a,b 〉 denotes the Hermitian product of two elements a and b in C N (the dimension is implicit). 1. A recursive decomposition, consequences 1.1. The general equality in law. Let G be a subgroup of U(N), the group of unitary matrices of size N. Let (e1,...,eN) be an orthonormal basis of C N and H: = {H ∈ G  H(e1) = e1}, the subgroup of G which
Law of log determinant of sample covariance matrix and optimal estimation of differential entropy for highdimensional gaussian distributions. arXiv:1309.0482
, 2013
"... Differential entropy and log determinant of the covariance matrix of a multivariate Gaussian distribution have many applications in coding, communications, signal processing and statistical inference. In this paper we consider in the high dimensional setting optimal estimation of the differential e ..."
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Differential entropy and log determinant of the covariance matrix of a multivariate Gaussian distribution have many applications in coding, communications, signal processing and statistical inference. In this paper we consider in the high dimensional setting optimal estimation of the differential entropy and the logdeterminant of the covariance matrix. We first establish a central limit theorem for the log determinant of the sample covariance matrix in the high dimensional setting where the dimension p(n) can grow with the sample size n. An estimator of the differential entropy and the log determinant is then considered. Optimal rate of convergence is obtained. It is shown that in the case p(n)/n → 0 the estimator is asymptotically sharp minimax. The ultrahigh dimensional setting where p(n)> n is also discussed.
LIMIT THEOREMS FOR ORTHOGONAL POLYNOMIALS RELATED TO CIRCULAR ENSEMBLES
, 2013
"... Abstract. For a natural extension of the circular unitary ensemble of order n, we study as n → ∞ the asymptotic behavior of the sequence of orthogonal polynomials with respect to the spectral measure. The last term of this sequence is the characteristic polynomial. After taking logarithm and rescal ..."
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Abstract. For a natural extension of the circular unitary ensemble of order n, we study as n → ∞ the asymptotic behavior of the sequence of orthogonal polynomials with respect to the spectral measure. The last term of this sequence is the characteristic polynomial. After taking logarithm and rescaling, we obtain a process indexed by t ∈ [0,1]. We show that it converges to a deterministic limit, and we describe the fluctuations and the large deviations. 1.
Operatorvalued spectral measures and large deviations
, 2013
"... Let H be a Hilbert space, U an unitary operator on H and K a cyclic subspace for U. The spectral measure of the pair (U,K) is an operatorvalued measure µK on the unit circle T such that T z k dµK(z) = PKU k) ↾K, ∀ k ≥ 0 where PK and ↾ K are the projection and restriction on K, respectively. When K ..."
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Let H be a Hilbert space, U an unitary operator on H and K a cyclic subspace for U. The spectral measure of the pair (U,K) is an operatorvalued measure µK on the unit circle T such that T z k dµK(z) = PKU k) ↾K, ∀ k ≥ 0 where PK and ↾ K are the projection and restriction on K, respectively. When K is one dimensional, µ is a scalar probability measure. In this case, if U is picked at random from the unitary group U(N) under the Haar measure, then any fixed K is almost surely cyclic for U. Let µ (N) be the random spectral (scalar) measure of (U,K). The sequence (µ (N) ) was studied extensively, in the regime of large N. It converges to the Haar measure λ on T and satisfies the Large Deviation Principle at scale N with a good rate function which is the reverse Kullback information with respect to λ ([20]). The purpose of the present paper is to give an extension of this result for general K (of fixed finite dimension p) and eventually to offer a projective statement (all p simultaneously), at the level of operatorvalued spectral measures in infinite dimensional spaces.