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Random matrix theory
, 2005
"... Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We includ ..."
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Cited by 82 (4 self)
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Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We include the important mathematics which is a very modern development, as well as the computational software that is transforming the theory into useful practice.
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.
Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence
 ANN. STATIST
, 2008
"... Let A and B be independent, central Wishart matrices in p variables with common covariance and having m and n degrees of freedom, respectively. The distribution of the largest eigenvalue of (A+B) −1 B has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppos ..."
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Cited by 30 (2 self)
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Let A and B be independent, central Wishart matrices in p variables with common covariance and having m and n degrees of freedom, respectively. The distribution of the largest eigenvalue of (A+B) −1 B has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that m and n grow in proportion to p. We show that after centering and scaling, the distribution is approximated to secondorder, O(p −2/3), by the Tracy–Widom law. The results are obtained for both complex and then realvalued data by using methods of random matrix theory to study the largest eigenvalue of the Jacobi unitary and orthogonal ensembles. Asymptotic approximations of Jacobi polynomials near the largest zero play a central role.
Tails of condition number distributions
 SIAM J. Matrix Anal. Appl
"... Abstract. Let κ be the condition number of an mbyn matrix with independent standard Gaussian entries, either real (β = 1) or complex (β = 2). The major result is the existence of a constant C (depending on m, n, β) such that P [κ> x] < C x −β for all x. As x → ∞, the bound is asymptotically ..."
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Cited by 28 (2 self)
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Abstract. Let κ be the condition number of an mbyn matrix with independent standard Gaussian entries, either real (β = 1) or complex (β = 2). The major result is the existence of a constant C (depending on m, n, β) such that P [κ> x] < C x −β for all x. As x → ∞, the bound is asymptotically tight. An analytic expression is given for the constant C, and simple estimates are given, one involving a TracyWidom largest eigenvalue distribution. All of the results extend beyond real and complex entries to general β.
Performance of statistical tests for singlesource detection using random matrix theory
 IEEE Transactions on Information Theory
, 2011
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MIMO broadcast scheduling with limited channel state information
 in Proc., Allerton Conf. on Comm., Control, and Computing
, 2005
"... We consider the multipleinput multipleoutput (MIMO) broadcast channel in which there arem transmit antennas and n uncoordinated users with a single receive antenna. We examine the maximum throughput in such a system in the scenario when the number of users is much greater than the number of trans ..."
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Cited by 19 (1 self)
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We consider the multipleinput multipleoutput (MIMO) broadcast channel in which there arem transmit antennas and n uncoordinated users with a single receive antenna. We examine the maximum throughput in such a system in the scenario when the number of users is much greater than the number of transmit antennas. We derive a lower bound the probability that there exists a set of users such that each user receives a specified rate. We show that through the use of a simple normthreshold feedback protocol the maximal scaling of the sum rate is achievable and requires channel state information (CSI) about only O(1) users. 1
Holonomic Gradient Method for the Distribution Function of the Largest Root of a Wishart Matrix
, 2012
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Accurate and efficient evaluation of Schur and Jack functions
 Math. Comp
, 2006
"... Abstract. We present new algorithms for computing the values of the Schur sλ(x1,x2,...,xn)andJackJ α λ (x1,x2,...,xn) functions in floating point arithmetic. These algorithms deliver guaranteed high relative accuracy for positive data (xi,α>0) and run in time that is only linear in n. 1. ..."
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Cited by 13 (4 self)
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Abstract. We present new algorithms for computing the values of the Schur sλ(x1,x2,...,xn)andJackJ α λ (x1,x2,...,xn) functions in floating point arithmetic. These algorithms deliver guaranteed high relative accuracy for positive data (xi,α>0) and run in time that is only linear in n. 1.
Accuracy of the Tracy–Widom limits for the extreme eigenvalues in white Wishart matrices
 BERNOULLI
, 2012
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