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44
On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices
, 1995
"... this paper by H. However, the assumptions on X share a common intersection: the entries of ..."
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Cited by 163 (14 self)
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this paper by H. However, the assumptions on X share a common intersection: the entries of
Eigenvalues of large sample covariance matrices of spiked population models
, 2006
"... We consider a spiked population model, proposed by Johnstone, whose population eigenvalues are all unit except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the nonunit population ones when both sample size and population size become large. This pape ..."
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Cited by 81 (5 self)
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We consider a spiked population model, proposed by Johnstone, whose population eigenvalues are all unit except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the nonunit population ones when both sample size and population size become large. This paper completely determines the almost sure limits for a general class of samples. 1
HighSNR power offset in multiantenna communication
 IEEE Transactions on Information Theory
, 2005
"... Abstract—The analysis of the multipleantenna capacity in the high regime has hitherto focused on the high slope (or maximum multiplexing gain), which quantifies the multiplicative increase as a function of the number of antennas. This traditional characterization is unable to assess the impact of ..."
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Cited by 59 (13 self)
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Abstract—The analysis of the multipleantenna capacity in the high regime has hitherto focused on the high slope (or maximum multiplexing gain), which quantifies the multiplicative increase as a function of the number of antennas. This traditional characterization is unable to assess the impact of prominent channel features since, for a majority of channels, the slope equals the minimum of the number of transmit and receive antennas. Furthermore, a characterization based solely on the slope captures only the scaling but it has no notion of the power required for a certain capacity. This paper advocates a more refined characterization whereby, as a function of �f, the high capacity is expanded as an affine function where the impact of channel features such as antenna correlation, unfaded components, etc., resides in the zeroorder term or power offset. The power offset, for which we find insightful closedform expressions, is shown to play a chief role for levels of practical interest. Index Terms—Antenna correlation, channel capacity, coherent communication, fading channels, high analysis, multiantenna arrays, Ricean channels.
Analysis of the Limiting Spectral Distribution of Large Dimensional Random Matrices
, 1995
"... this paper is to derive certain fundamental properties, the most important being the analyticity of F . Under condition 2) it is shown in [8] that F 0 has moments of all order satisfying Carleman's su#ciency condition, and are explicitly expressed. From the moments, F 0 ..."
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Cited by 44 (11 self)
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this paper is to derive certain fundamental properties, the most important being the analyticity of F . Under condition 2) it is shown in [8] that F 0 has moments of all order satisfying Carleman's su#ciency condition, and are explicitly expressed. From the moments, F 0
CLT for Linear Spectral Statistics of Large Dimensional Sample Covariance Matrices
, 2003
"... This paper shows their of rate of convergence to be 1/n by proving, after proper scaling, they form a tight sequence. Moreover, if EX 11 =0andEX11 =2, or if X11 and T n are real and EX 11 = 3, they are shown to have Gaussian limits ..."
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Cited by 37 (0 self)
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This paper shows their of rate of convergence to be 1/n by proving, after proper scaling, they form a tight sequence. Moreover, if EX 11 =0andEX11 =2, or if X11 and T n are real and EX 11 = 3, they are shown to have Gaussian limits
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.
SPECTRUM ESTIMATION FOR LARGE DIMENSIONAL COVARIANCE MATRICES USING RANDOM MATRIX THEORY
 SUBMITTED TO THE ANNALS OF STATISTICS
"... Estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics; the eigenvalues of covariance matrices play a key role in many widely techniques, in particular in Principal Component Analysis (PCA). In ma ..."
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Cited by 27 (4 self)
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Estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics; the eigenvalues of covariance matrices play a key role in many widely techniques, in particular in Principal Component Analysis (PCA). In many modern data analysis problems, statisticians are faced with large datasets where the sample size, n, is of the same order of magnitude as the number of variables p. Random matrix theory predicts that in this context, the eigenvalues of the sample covariance matrix are not good estimators of the eigenvalues of the population covariance. We propose to use a fundamental result in random matrix theory, the MarčenkoPastur equation, to better estimate the eigenvalues of large dimensional covariance matrices. The MarčenkoPastur equation holds in very wide generality and under weak assumptions. The estimator we obtain can be thought of as “shrinking ” in a non linear fashion the eigenvalues of the sample covariance matrix to estimate the population eigenvalues. Inspired by ideas of random matrix theory, we also suggest a change of point of view when thinking about estimation of highdimensional vectors: we do not try to estimate directly the vectors but rather a probability measure that describes them. We think this is a theoretically more fruitful way to think about these problems. Our estimator gives fast and good or very good results in extended simulations. Our algorithmic approach is based on convex optimization. We also show that the proposed estimator is consistent.
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
Design of reducedrank MMSE multiuser detectors using random matrix methods
 IEEE Trans. Inform. Theory
, 2004
"... Abstract—Reducedrank minimum meansquared error (MMSE) multiuser detectors using asymptotic weights have been shown to reduce receiver complexity while maintaining good performance in longsequence codedivision multipleaccess (CDMA) systems. In this paper, we consider the design of reducedrank M ..."
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Cited by 21 (1 self)
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Abstract—Reducedrank minimum meansquared error (MMSE) multiuser detectors using asymptotic weights have been shown to reduce receiver complexity while maintaining good performance in longsequence codedivision multipleaccess (CDMA) systems. In this paper, we consider the design of reducedrank MMSE receivers in a general framework which includes fading, single and multiantenna receivers, as well as directsequence CDMA (DSCDMA) and multicarrier CDMA (both uplink and downlink). In all these cases, random matrix results are used to obtain explicit expressions for the asymptotic eigenvalue moments of the interference autocorrelation matrix and for the asymptotic weights used in the reducedrank receiver. Index Terms—Codedivision multiple access (CDMA), fading channels, minimum meansquared error (MMSE) receivers, multiantenna systems, multicarrier CDMA, multiuser detection, random matrix theory. I.
Random matrices: The distribution of the smallest singular values
, 2009
"... Let ξ be a realvalued random variable of mean zero and variance 1. Let Mn(ξ) denote the n × n random matrix whose entries are iid copies of ξ and σn(Mn(ξ)) denote the least singular value of Mn(ξ). The quantity σn(Mn(ξ)) 2 is thus the least eigenvalue of the Wishart matrix MnM ∗ n. We show that ( ..."
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Cited by 20 (1 self)
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Let ξ be a realvalued random variable of mean zero and variance 1. Let Mn(ξ) denote the n × n random matrix whose entries are iid copies of ξ and σn(Mn(ξ)) denote the least singular value of Mn(ξ). The quantity σn(Mn(ξ)) 2 is thus the least eigenvalue of the Wishart matrix MnM ∗ n. We show that (under a finite moment assumption) the probability distribution nσn(Mn(ξ)) 2 is universal in the sense that it does not depend on the distribution of ξ. In particular, it converges to the same limiting distribution as in the special case when ξ is real gaussian. (The limiting distribution was computed explicitly in this case by Edelman.) We also proved a similar result for complexvalued random variables of mean zero, with real and imaginary parts having variance 1/2 and covariance zero. Similar results are also obtained for the joint distribution of the bottom k singular values of Mn(ξ) for any fixed k (or even for k growing as a small power of n) and for rectangular matrices. Our approach is motivated by the general idea of “property testing ” from combinatorics and theoretical computer science. This seems to be a new approach in the study of spectra of random matrices and combines tools from various areas of mathematics