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Second order freeness and Fluctuations of Random Matrices : I. Gaussian and Wishart matrices and cyclic Fock spaces
, 2005
"... We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of “second order freeness” and interpret the global fluctuations of Gaussian and Wishart random matrices by a general limit theorem for s ..."
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Cited by 52 (6 self)
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We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of “second order freeness” and interpret the global fluctuations of Gaussian and Wishart random matrices by a general limit theorem for second order freeness. By introducing cyclic Fock space, we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation, annihilation, and preservation operators. We show that orthogonal families of Gaussian and Wishart random matrices are asymptotically free of second order.
Annular noncrossing permutations and partitions, and secondorder asymptotics for random matrices
, 2003
"... ..."
All invariant moments of the Wishart distribution
, 2004
"... In this paper, we compute moments of a Wishart matrix variate U of the form E(Q(U)) where Q(u) is a polynomial with respect to the entries of the symmetric matrix u, invariant in the sense that it depends only on the eigenvalues of the matrix u. This gives us in particular the expectedvalue of any ..."
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Cited by 17 (1 self)
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In this paper, we compute moments of a Wishart matrix variate U of the form E(Q(U)) where Q(u) is a polynomial with respect to the entries of the symmetric matrix u, invariant in the sense that it depends only on the eigenvalues of the matrix u. This gives us in particular the expectedvalue of any power of the Wishart matrix U or its inverse U1. For our proofs, we do not rely on traditional combinatorial methods but rather on the interplay between two bases of the space of invariant polynomials in U. This means that all moments can be obtainedthrough the multiplication of three matrices with known entries. Practically, the moments are obtainedby computer with an extremely simple Maple program.
Gaussianization and eigenvalue statistics for random quantum channels (III)
 ANN. APPL. PROBAB
, 2009
"... In this paper, we present applications of the calculus developed in [9], and obtain an exact formula for the moments of random quantum channels whose input is a pure state thanks to gaussianization methods. Our main application is an indepth study of the random matrix model introduced by Hayden a ..."
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Cited by 15 (9 self)
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In this paper, we present applications of the calculus developed in [9], and obtain an exact formula for the moments of random quantum channels whose input is a pure state thanks to gaussianization methods. Our main application is an indepth study of the random matrix model introduced by Hayden and Winter and used recently by Brandao, Horodecki, Fukuda and King to refine the Hastings counterexample to the additivity conjecture in Quantum Information Theory. This model is exotic from the point of view of random matrix theory, as its eigenvalues obey to two different scalings simultaneously. We study its asymptotic behavior and obtain an asymptotic expansion for its von Neumann entropy.
On the Capacity of MIMO Wireless Channels with Dynamic CSIT
 IEEE JOURNAL ON SELECTED AREAS IN COMM., SPECIAL
, 2007
"... Transmit channel side information (CSIT) can significantly increase MIMO wireless capacity. Due to delay in acquiring this information, however, the timeselective fading wireless channel often induces incomplete, or partial, CSIT. In this paper, we first construct a dynamic CSIT model that takes i ..."
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Cited by 12 (3 self)
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Transmit channel side information (CSIT) can significantly increase MIMO wireless capacity. Due to delay in acquiring this information, however, the timeselective fading wireless channel often induces incomplete, or partial, CSIT. In this paper, we first construct a dynamic CSIT model that takes into account channel temporal variation. It does so by using a potentially outdated channel measurement and the channel statistics, including the mean, covariance, and temporal correlation. The dynamic CSIT model consists of an effective channel mean and an effective channel covariance, derived as a channel estimate and its error covariance. Both parameters are functions of the temporal correlation factor, which indicates the CSIT quality. Depending on this quality, the model covers smoothly from perfect to statistical CSIT. We then summarize and further analyze the capacity gains and
Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss
 Journal of Multivariate Analysis
, 2009
"... complex normal distributions under an invariant quadratic loss ..."
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Cited by 4 (0 self)
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complex normal distributions under an invariant quadratic loss
Compound real Wishart and qWishart matrices
, 2007
"... We introduce a family of matrices with noncommutative entries that generalize the classical real Wishart matrices. With the help of the Brauer product, we derive a nonasymptotic expression for the moments of traces of monomials in such matrices; the expression is quite similar to the formula deriv ..."
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Cited by 3 (0 self)
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We introduce a family of matrices with noncommutative entries that generalize the classical real Wishart matrices. With the help of the Brauer product, we derive a nonasymptotic expression for the moments of traces of monomials in such matrices; the expression is quite similar to the formula derived in [9, Theorem 2.1] for independent complex Wishart matrices. We then analyze the fluctuations about the MarchenkoPastur law. We show that after centering by the mean, traces of real symmetric polynomials in qWishart matrices converge in distribution, and we identify the asymptotic law as the normal law when q = 1, and as the semicircle law when q = 0.