Results 1  10
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37
Concentration of the Spectral Measure for Large Matrices
, 2000
"... We derive concentration inequalities for functions of the empirical measure of eigenvalues for large, random, self adjoint matrices, with not necessarily Gaussian entries. The results presented apply in particular to nonGaussian Wigner and Wishart matrices. We also provide concentration bounds for ..."
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Cited by 65 (11 self)
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We derive concentration inequalities for functions of the empirical measure of eigenvalues for large, random, self adjoint matrices, with not necessarily Gaussian entries. The results presented apply in particular to nonGaussian Wigner and Wishart matrices. We also provide concentration bounds for non commutative functionals of random matrices. 1 Introduction and statement of results Consider a random N N Hermitian matrix X with i.i.d. complex entries (except for the symmetry constraint) satisfying a moment condition. It is well known since Wigner [28] that the spectral measure of N 1=2 X converges to the semicircle law. This observation has been generalized to a large class of matrices, e.g. sample covariance matrices of the form XRX where R is a deterministic diagonal matrix ([19]), band matrices (see [5, 16, 20]), etc. For the Wigner case, this convergence has been supplemented by Central Limit Theorems, see [15] for the case of Gaussian entries and [17], [22] for the gen...
Capacity of multiantenna array systems in indoor wireless environment
 IEEE Globecom
, 1998
"... Studies show that multipleelement antenna arrays (MEA) with n transmitters and n receivers can achieve n more bits/Hz than singleantenna systems in an independent Rayleigh fading environment. In this paper, we explore the behavior of MEA capacities in a more realistic propagation environment simul ..."
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Cited by 36 (3 self)
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Studies show that multipleelement antenna arrays (MEA) with n transmitters and n receivers can achieve n more bits/Hz than singleantenna systems in an independent Rayleigh fading environment. In this paper, we explore the behavior of MEA capacities in a more realistic propagation environment simulated via the WiSE ray tracing tool. We impose an average power constraint and collect statistics of the capacity, Cwf and mutual information Ieq. In addition, we derive mathematically the asymptotic growth rates Cwf /n and Ieq /n as n → ∞ for two cases: (a) independent fadings and (b) spatially correlated fadings between antennas. Cwf /n and Ieq /n converge to constants Cwf * and Ieq *, respectively in case (a), o o o o and to Cwf and Ieq in case (b). Cwf and Ieq predict very closely the slope observed in simulations, even at moderate n = 16. I.
The central limit theorem for local linear s tatistics in classical compact groups and related combinatorial identities
 Ann. Probab
, 2000
"... We discuss CLT for the global and local linear statistics of random matrices from classical compact groups. The main part of our proofs are certain combinatorial identities much in the spirit of works by Kac and Spohn. 1 ..."
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Cited by 27 (2 self)
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We discuss CLT for the global and local linear statistics of random matrices from classical compact groups. The main part of our proofs are certain combinatorial identities much in the spirit of works by Kac and Spohn. 1
DETERMINISTIC EQUIVALENTS FOR CERTAIN FUNCTIONALS OF LARGE RANDOM MATRICES 1
, 2005
"... Consider an N × n random matrix Yn = (Y n ij) where the entries are given by Y n ij = σij(n) √ X n n ij, the X n ij being independent and identically distributed, centered with unit variance and satisfying some mild moment assumption. Consider now a deterministic N ×n matrix An whose columns and row ..."
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Cited by 26 (14 self)
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Consider an N × n random matrix Yn = (Y n ij) where the entries are given by Y n ij = σij(n) √ X n n ij, the X n ij being independent and identically distributed, centered with unit variance and satisfying some mild moment assumption. Consider now a deterministic N ×n matrix An whose columns and rows are uniformly bounded in the Euclidean norm. Let Σn = Yn + An. We prove in this article that there exists a deterministic N ×N matrixvalued function Tn(z) analytic in C −R + such that, almost surely, 1 lim
Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields
, 2008
"... Abstract. Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a ..."
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Cited by 24 (3 self)
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Abstract. Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of ‘second order Poincaré inequalities’: just as ordinary Poincaré inequalities give variance bounds, second order Poincaré inequalities give central limit theorems. The proof of the main result employs Stein’s method of normal approximation. A number of examples are worked out; some of them are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices. 1.
The largest eigenvalue of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations
, 2007
"... ..."
A CLT for a band matrix model
 Probab. Theory Relat. Fields
, 2005
"... Abstract. A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose onorabove diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on system ..."
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Cited by 15 (0 self)
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Abstract. A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose onorabove diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on systematic combinatorial enumeration, study of generating functions, and concentration inequalities of the Poincaré type. Special cases treated, with an explicit evaluation of limiting variances, are generalized Wigner and Wishart matrices. 1.
Large deviations and stochastic calculus for large random matrices
, 2004
"... Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of math ..."
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Cited by 13 (0 self)
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Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of mathematical breakthroughs allowing for instance a better understanding of local properties of their spectrum, answering universality questions, connecting these issues with growth processes etc. In this survey, we shall discuss the problem of the large deviations of the empirical measure of Gaussian random matrices, and more generally of the trace of words of independent Gaussian random matrices. We shall describe how such issues are motivated either in physics/combinatorics by the study of the socalled matrix models or in free probability by the definition of a noncommutative entropy. We shall show how classical large deviations techniques can be used in this context. These lecture notes are supposed to be accessible to non probabilists and non freeprobabilists.
Gaussian fluctuation for the number of particles in Airy, Bessel, sine, and other determinantal random point
, 2000
"... We prove the Central Limit Theorem for the number of eigenvalues near the spectrum edge for hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the Gaussian fluctuation of the number of particles in random point ..."
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Cited by 13 (1 self)
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We prove the Central Limit Theorem for the number of eigenvalues near the spectrum edge for hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the Gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions. As another corollary of CostinLebowitz Theorem we prove CLT for the empirical distribution function of the eigenvalues of random matrices from classical compact groups. 1 Introduction and Formulation of Results Random hermitian matrices were introduced in mathematical physics by Wigner in the fifties ([1], [2]). The main motivation of pioneers in this field 1 was to obtain a better understanding of the statistical behavior of energy levels of heavy nuclei. An archetypical example of random matrices is the
Capacity Scaling in DualAntennaArray Wireless Systems
 IEEE TRANS. ON COMMUN
, 2000
"... Wireless systems using multielement antenna arrays simultaneously at both transmitter and receiver promise a much higher capacity than conventional systems. Previous studies have shown that singleuser systems employing nelement transmit and receive arrays can achieve a capacity proportional to ..."
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Cited by 12 (0 self)
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Wireless systems using multielement antenna arrays simultaneously at both transmitter and receiver promise a much higher capacity than conventional systems. Previous studies have shown that singleuser systems employing nelement transmit and receive arrays can achieve a capacity proportional to n, assuming independent Rayleigh fading between pairs of antenna elements. We explore the capacity of dualantennaarray systems via theoretical analysis and simulation experiments, focusing particularly on the situation when there is correlation between the fading at different antennas. We derive and compare expressions for the asymptotic growth rate of capacity with n antennas for both independent and correlated fading cases; the latter is derived under some assumptions about the scaling of the fading correlation structure. We show that the capacity growth is linear in n in both the independent and correlated cases, but the growth rate is smaller in the latter case. We compare the ...