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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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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.
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
On the largest singular values of random matrices with independent Cauchy entries
 J. MATH. PHYS
, 2005
"... We apply the method of determinants to study the distribution of the largest singular values of large m × n real rectangular random matrices with independent Cauchy entries. We show that statistical properties of the (rescaled by a factor 1 m 2 n 2) largest singular values agree in the limit with th ..."
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We apply the method of determinants to study the distribution of the largest singular values of large m × n real rectangular random matrices with independent Cauchy entries. We show that statistical properties of the (rescaled by a factor 1 m 2 n 2) largest singular values agree in the limit with the statistics of the Poisson random point process with the intensity 1 π x−3/2 and, therefore, are different from the TracyWidom law. Among other corollaries of our method we show an interesting connection between the mathematical expectations of the determinants of the complex rectangular m×n standard Wishart ensemble and the real rectangular 2m×2n standard Wishart ensemble.
Spectral measure of heavy tailed band and covariance random matrices, preprintarXiv:0811.1587v2 [math.PR
 Mathematical Physics
"... Abstract. We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure ˆµ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N × N symmetric matrix Yσ i j N whose (i, j) entry is σ ( , N N)xij where (xij,1 ≤ i ≤ j < ..."
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Abstract. We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure ˆµ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N × N symmetric matrix Yσ i j N whose (i, j) entry is σ ( , N N)xij where (xij,1 ≤ i ≤ j < ∞) is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an αstable law, α ∈ (0, 2), and σ is a deterministic function. For random diagonal DN independent of Yσ N and with appropriate rescaling aN, we prove that ˆµ −1 a N Yσ N +D converges in mean towards a limiting probability N measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries. 1.
Poisson convergence for the largest eigenvalues of Heavy Tailed Random Matrices
, 2008
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The spectrum of heavy tailed random matrices
, 2007
"... Let XN be an N × N random symmetric matrix with independent equidistributed entries modulo the symmetry constraint. If the law P of the entries has a finite second moment, it was shown by Wigner [14] that the empirical distribution of the eigenvalues of XN, once renormalized by √ N, converges almost ..."
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Let XN be an N × N random symmetric matrix with independent equidistributed entries modulo the symmetry constraint. If the law P of the entries has a finite second moment, it was shown by Wigner [14] that the empirical distribution of the eigenvalues of XN, once renormalized by √ N, converges almost surely and in expectation to the socalled semicircular distribution as N goes to infinity. In this paper we study the same question when P is in the domain of attraction of an αstable law. We prove that if we renormalize the eigenvalues by a constant aN of order N 1 α, the corresponding spectral distribution converges in expectation towards a law µα which only depends on α. We characterize µα and study some of its properties; it is a heavytailed probability measure which is absolutely continuous with respect to Lebesgue measure except possibly on a compact set of capacity zero. 1
LOCALIZATION AND DELOCALIZATION FOR HEAVY TAILED BAND MATRICES
"... Abstract. We consider some random band matrices with bandwidth N µ whose entries are independent random variables with distribution tail in x −α. We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when α < 2(1 + µ −1), t ..."
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Abstract. We consider some random band matrices with bandwidth N µ whose entries are independent random variables with distribution tail in x −α. We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when α < 2(1 + µ −1), the largest eigenvalues have order N 1+µ α, are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked by Soshnikov in [29, 30] for full matrices with heavy tailed entries, i.e. when α < 2, and by Auffinger et al in [1] when α < 4). On the other hand, when α> 2(1 + µ −1), the largest eigenvalues have order N µ 2 and most eigenvectors of the matrix are delocalized, i.e. approximately uniformly distributed on their N coordinates.
Scaled Limit and Rate of Convergence for the Largest Eigenvalue from the Generalized Cauchy Random Matrix Ensemble
, 2009
"... Abstract. In this paper, we are interested in the asymptotic properties for the largest eigenvalue of the Hermitian random matrix ensemble, called the Generalized Cauchy ensemble GCy, whose eigenvalues PDF is given by const · (xj − xk) 2 ..."
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Abstract. In this paper, we are interested in the asymptotic properties for the largest eigenvalue of the Hermitian random matrix ensemble, called the Generalized Cauchy ensemble GCy, whose eigenvalues PDF is given by const · (xj − xk) 2
Heavy tails in lastpassage percolation
, 2005
"... We consider lastpassage percolation models in two dimensions, in which the underlying weight distribution has a heavy tail of index α < 2. We prove scaling laws and asymptotic distributions, both for the passage times and for the shape of optimal paths; these are expressed in terms of a family ( ..."
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We consider lastpassage percolation models in two dimensions, in which the underlying weight distribution has a heavy tail of index α < 2. We prove scaling laws and asymptotic distributions, both for the passage times and for the shape of optimal paths; these are expressed in terms of a family (indexed by α) of “continuous lastpassage percolation ” models in the unit square. In the extreme case α = 0 (corresponding to a distribution with slowly varying tail) the asymptotic distribution of the optimal path can be represented by a random selfsimilar measure on [0,1], whose multifractal spectrum we compute. By extending the continuous lastpassage percolation model to R 2 we obtain a heavytailed analogue of the Airy process, representing the limit of appropriately scaled vectors of passage times to different points in the plane. We give corresponding results for a directed percolation problem based on αstable Lévy processes, and indicate extensions of the results to higher dimensions. 1