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29
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 inter-dependence. 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 inter-dependence. 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.
Universality for certain hermitian Wigner matrices under weak moment conditions
, 2011
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Poisson convergence for the largest eigenvalues of Heavy Tailed Random Matrices
, 2008
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A generalization of Wigner’s law
- Comm. Math. Phys
"... We present a generalization of Wigner's semicircle law: we con-sider 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 con-sider 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 k-th 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
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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Cited by 14 (1 self)
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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 α-stable 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.
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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Cited by 10 (1 self)
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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 so-called 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 α-stable 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 heavy-tailed probability measure which is absolutely continuous with respect to Lebesgue measure except possibly on a compact set of capacity zero. 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 Tracy-Widom 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.
