@article{zhou_hu_bai_silverstein_2023, title={Analysis of the Limiting Spectral Distribution of Large-dimensional General Information-Plus-Noise-Type Matrices}, volume={7}, ISSN={["1572-9230"]}, url={https://doi.org/10.1007/s10959-023-01276-3}, DOI={10.1007/s10959-023-01276-3}, abstractNote={In this paper, we derive the analytical behavior of the limiting spectral distribution of non-central covariance matrices of the “general information-plus-noise" type, as studied in Zhou (JTP 36:1203–1226, 2023). Through the equation defining its Stieltjes transform, it is shown that the limiting distribution has a continuous derivative away from zero, the derivative being analytic wherever it is positive, and we show the determination criterion for its support. We also extend the result in Zhou (JTP 36:1203-1226, 2023) to allow for all possible ratios of row to column of the underlying random matrix.}, journal={JOURNAL OF THEORETICAL PROBABILITY}, author={Zhou, Huanchao and Hu, Jiang and Bai, Zhidong and Silverstein, Jack W.}, year={2023}, month={Jul} }
 @article{bai_silverstein_2022, title={A tribute to PR Krishnaiah}, volume={188}, ISSN={["0047-259X"]}, url={https://doi.org/10.1016/j.jmva.2021.104828}, DOI={10.1016/j.jmva.2021.104828}, abstractNote={The authors reminisce on their association with P.R. Krishnaiah, renowned professor of statistics at the University of Pittsburgh and founding editor of the Journal of Multivariate Analysis. They recount their individual associations with him, mainly involving the behavior of eigenvalues of random matrices, and outline two areas of applied work he performed with one of the authors.}, journal={JOURNAL OF MULTIVARIATE ANALYSIS}, publisher={Elsevier BV}, author={Bai, Zhidong and Silverstein, Jack W.}, year={2022}, month={Mar} }
 @article{silverstein_2022, title={Limiting eigenvalue behavior of a class of large dimensional random matrices formed from a Hadamard product}, volume={7}, ISSN={["2010-3271"]}, DOI={10.1142/S2010326322500502}, abstractNote={ This paper investigates the strong limiting behavior of the eigenvalues of the class of matrices [Formula: see text], studied in [V. L. Girko, Theory of Stochastic Canonical Equations: Vol. [Formula: see text] (Kluwer Academic Publishers, Dordrecht, 2001)]. Here, [Formula: see text] is an [Formula: see text] random matrix consisting of independent complex standardized random variables, [Formula: see text], [Formula: see text], has nonnegative entries, and ∘ denotes Hadamard (componentwise) product. Results are obtained under assumptions on the entries of [Formula: see text] and [Formula: see text] which are different from those in [V. L. Girko, Theory of Stochastic Canonical Equations: Vol. 1 (Kluwer Academic Publishers, Dordrecht, 2001)], which include a Lindeberg condition on the entries of [Formula: see text], as well as a bound on the average of the rows and columns of [Formula: see text]. The present paper separates the assumptions needed on [Formula: see text] and [Formula: see text]. It assumes a Lindeberg condition on the entries of [Formula: see text], along with a tightness-like condition on the entries of [Formula: see text]. }, journal={RANDOM MATRICES-THEORY AND APPLICATIONS}, author={Silverstein, Jack W.}, year={2022}, month={Jul} }
 @article{silverstein_2022, title={Weak convergence of a collection of random functions defined by the eigenvectors of large dimensional random matrices}, volume={11}, ISSN={["2010-3271"]}, url={https://doi.org/10.1142/S2010326322500332}, DOI={10.1142/S2010326322500332}, abstractNote={ For each n, let [Formula: see text] be Haar distributed on the group of [Formula: see text] unitary matrices. Let [Formula: see text] denote orthogonal nonrandom unit vectors in [Formula: see text] and let [Formula: see text], [Formula: see text]. Define the following functions on [Formula: see text]: [Formula: see text], [Formula: see text], [Formula: see text]. Then it is proven that [Formula: see text], [Formula: see text], considered as random processes in [Formula: see text], converge weakly, as [Formula: see text], to [Formula: see text] independent copies of Brownian bridge. The same result holds for the [Formula: see text] processes in the real case, where [Formula: see text] is real orthogonal Haar distributed and [Formula: see text], with [Formula: see text] in [Formula: see text] and [Formula: see text] in [Formula: see text] replaced with [Formula: see text] and [Formula: see text], respectively. This latter result will be shown to hold for the matrix of eigenvectors of [Formula: see text] where [Formula: see text] is [Formula: see text] consisting of the entries of [Formula: see text], i.i.d. standardized and symmetrically distributed, with each [Formula: see text] and [Formula: see text] as [Formula: see text]. This result extends the result in [J. W. Silverstein, Ann. Probab. 18 (1990) 1174–1194]. These results are applied to the detection problem in sampling random vectors mostly made of noise and detecting whether the sample includes a nonrandom vector. The matrix [Formula: see text] is studied where [Formula: see text] is Hermitian or symmetric and nonnegative definite with either its matrix of eigenvectors being Haar distributed, or [Formula: see text], [Formula: see text] nonrandom and [Formula: see text] is a nonrandom unit vector. Results are derived on the distributional behavior of the inner product of vectors orthogonal to [Formula: see text] with the eigenvector associated with the largest eigenvalue of [Formula: see text] }, number={04}, journal={RANDOM MATRICES-THEORY AND APPLICATIONS}, publisher={World Scientific Pub Co Pte Ltd}, author={Silverstein, Jack W.}, year={2022}, month={Oct} }
 @article{local convergence of an amp variant to the lasso solution in finite dimensions_2020, year={2020}, month={Jul} }
 @article{weak convergence of a collection of random functions defined by the eigenvectors of large dimensional random matrices_2020, year={2020}, month={Dec} }
 @article{bryc_silverstein_2019, title={Singular values of large non-central random matrices}, volume={6}, ISSN={2010-3263 2010-3271}, url={http://dx.doi.org/10.1142/s2010326320500124}, DOI={10.1142/s2010326320500124}, abstractNote={ We study largest singular values of large random matrices, each with mean of a fixed rank [Formula: see text]. Our main result is a limit theorem as the number of rows and columns approach infinity, while their ratio approaches a positive constant. It provides a decomposition of the largest [Formula: see text] singular values into the deterministic rate of growth, random centered fluctuations given as explicit linear combinations of the entries of the matrix, and a term negligible in probability. We use this representation to establish asymptotic normality of the largest singular values for random matrices with means that have block structure. We also deduce asymptotic normality for the largest eigenvalues of a random matrix arising in a model of population genetics. }, journal={Random Matrices: Theory and Applications}, publisher={World Scientific Pub Co Pte Lt}, author={Bryc, Włodek and Silverstein, Jack W.}, year={2019}, month={Jun}, pages={2050012} }
 @article{couillet_pascal_silverstein_2015, title={The random matrix regime of Maronna's M-estimator with elliptically distributed samples}, volume={139}, ISSN={["0047-259X"]}, DOI={10.1016/j.jmva.2015.02.020}, abstractNote={This article demonstrates that the robust scatter matrix estimator CˆN∈CN×N of a multivariate elliptical population x1,…,xn∈CN originally proposed by Maronna in 1976, and defined as the solution (when existent) of an implicit equation, behaves similar to a well-known random matrix model in the limiting regime where the population N and sample n sizes grow at the same speed. We show precisely that CˆN∈CN×N is defined for all n large with probability one and that, under some light hypotheses, ‖CˆN−SˆN‖→0 almost surely in spectral norm, where SˆN follows a classical random matrix model. As a corollary, the limiting eigenvalue distribution of CˆN is derived. This analysis finds applications in the fields of statistical inference and signal processing.}, journal={JOURNAL OF MULTIVARIATE ANALYSIS}, author={Couillet, Romain and Pascal, Frederic and Silverstein, Jack W.}, year={2015}, month={Jul}, pages={56–78} }
 @article{wang_silverstein_yao_2014, title={A note on the CLT of the LSS for sample covariance matrix from a spiked population model}, volume={130}, ISSN={["0047-259X"]}, DOI={10.1016/j.jmva.2014.04.021}, abstractNote={In this note, we establish an asymptotic expansion for the centering parameter appearing in the central limit theorems for linear spectral statistic of large-dimensional sample covariance matrices when the population has a spiked covariance structure. As an application, we provide an asymptotic power function for the corrected likelihood ratio statistic for testing the presence of spike eigenvalues in the population covariance matrix. This result generalizes an existing formula from the literature where only one simple spike exists.}, journal={JOURNAL OF MULTIVARIATE ANALYSIS}, author={Wang, Qinwen and Silverstein, Jack W. and Yao, Jian-Feng}, year={2014}, month={Sep}, pages={194–207} }
 @article{couillet_pascal_silverstein_2014, title={Robust Estimates of Covariance Matrices in the Large Dimensional Regime}, volume={60}, ISSN={["1557-9654"]}, DOI={10.1109/tit.2014.2354045}, abstractNote={This paper studies the limiting behavior of a class of robust population covariance matrix estimators, originally due to Maronna in 1976, in the regime where both the number of available samples and the population size grow large. Using tools from random matrix theory, we prove that, for sample vectors made of independent entries having some moment conditions, the difference between the sample covariance matrix and (a scaled version of) such robust estimator tends to zero in spectral norm, almost surely. This result can be applied to various statistical methods arising from random matrix theory that can be made robust without altering their first order behavior.}, number={11}, journal={IEEE TRANSACTIONS ON INFORMATION THEORY}, author={Couillet, Romain and Pascal, Frederic and Silverstein, Jack W.}, year={2014}, month={Nov}, pages={7269–7278} }
 @inproceedings{couillet_pascal_silverstein_2013, title={A joint robust estimation and random matrix framework with application to array processing}, ISBN={9781479903566}, url={http://dx.doi.org/10.1109/icassp.2013.6638930}, DOI={10.1109/icassp.2013.6638930}, abstractNote={An original interface between robust estimation theory and random matrix theory for the estimation of population covariance matrices is proposed. Consider a random vector x = A<sub>N</sub>y ∈ C<sup>N</sup> with y ∈ C<sup>M</sup> made of M ≥ N independent entries, E[y] = 0, and E[yy*] = I<sub>N</sub>. It is shown that a class of robust estimators Ĉ<sub>N</sub> of C<sub>N</sub> = A<sub>N</sub>A*<sub>N</sub>, obtained from n independent copies of x, is (N, n)-consistent with the traditional sample covariance matrix r̂<sub>N</sub> in the sense that ∥Ĉ<sub>N</sub> - αr̂<sub>N</sub>∥ → 0 in spectral norm for some α > 0, almost surely, as N, n → ∞ with N/n and M/N bounded. This result, in general not valid in the fixed N regime, is used to propose improved subspace estimation techniques, among which an enhanced direction-of-arrival estimator called robust G-MUSIC.}, booktitle={2013 IEEE International Conference on Acoustics, Speech and Signal Processing}, publisher={IEEE}, author={Couillet, Romain and Pascal, Frederic and Silverstein, Jack W.}, year={2013}, month={May} }
 @article{bryc_bryc_silverstein_2013, title={Separation of the largest eigenvalues in eigenanalysis of genotype data from discrete subpopulations}, volume={89}, ISSN={["1096-0325"]}, DOI={10.1016/j.tpb.2013.08.004}, abstractNote={We present a mathematical model, and the corresponding mathematical analysis, that justifies and quantifies the use of principal component analysis of biallelic genetic marker data for a set of individuals to detect the number of subpopulations represented in the data. We indicate that the power of the technique relies more on the number of individuals genotyped than on the number of markers.}, journal={THEORETICAL POPULATION BIOLOGY}, author={Bryc, Katarzyna and Bryc, Wlodek and Silverstein, Jack W.}, year={2013}, month={Nov}, pages={34–43} }
 @article{hachem_kharouf_najim_silverstein_2012, title={A CLT FOR INFORMATION-THEORETIC STATISTICS OF NON-CENTERED GRAM RANDOM MATRICES}, volume={01}, ISSN={2010-3263 2010-3271}, url={http://dx.doi.org/10.1142/s2010326311500109}, DOI={10.1142/s2010326311500109}, abstractNote={ In this article, we study the fluctuations of the random variable: [Formula: see text] where [Formula: see text], as the dimensions of the matrices go to infinity at the same pace. Matrices Xn and An are respectively random and deterministic N × n matrices; matrices Dn and [Formula: see text] are deterministic and diagonal, with respective dimensions N × N and n × n; matrix Xn = (Xij) has centered, independent and identically distributed entries with unit variance, either real or complex. }, number={02}, journal={Random Matrices: Theory and Applications}, publisher={World Scientific Pub Co Pte Lt}, author={Hachem, Walid and Kharouf, Malika and Najim, Jamal and Silverstein, Jack W.}, year={2012}, month={Apr}, pages={1150010} }
 @article{bai_silverstein_2012, title={NO EIGENVALUES OUTSIDE THE SUPPORT OF THE LIMITING SPECTRAL DISTRIBUTION OF INFORMATION-PLUS-NOISE TYPE MATRICES}, volume={01}, ISSN={2010-3263 2010-3271}, url={http://dx.doi.org/10.1142/s2010326311500043}, DOI={10.1142/s2010326311500043}, abstractNote={ We consider a class of matrices of the form Cn = (1/N)(Rn + σXn)(Rn + σXn)*, where Xn is an n × N matrix consisting of independent standardized complex entries, Rn is an n × N nonrandom matrix, and σ > 0. Among several applications, Cn can be viewed as a sample correlation matrix, where information is contained in [Formula: see text], but each column of Rn is contaminated by noise. As n → ∞, if n/N → c > 0, and the empirical distribution of the eigenvalues of [Formula: see text] converge to a proper probability distribution, then the empirical distribution of the eigenvalues of Cn converges a.s. to a nonrandom limit. In this paper we show that, under certain conditions on Rn, for any closed interval in ℝ+ outside the support of the limiting distribution, then, almost surely, no eigenvalues of Cn will appear in this interval for all n large. }, number={01}, journal={Random Matrices: Theory and Applications}, publisher={World Scientific Pub Co Pte Lt}, author={Bai, Zhidong and Silverstein, Jack W.}, year={2012}, month={Jan}, pages={1150004} }
 @article{couillet_debbah_silverstein_2011, title={A Deterministic Equivalent for the Analysis of Correlated MIMO Multiple Access Channels}, volume={57}, ISSN={0018-9448 1557-9654}, url={http://dx.doi.org/10.1109/tit.2011.2133151}, DOI={10.1109/tit.2011.2133151}, abstractNote={In this article, novel deterministic equivalents for the Stieltjes transform and the Shannon transform of a class of large dimensional random matrices are provided. These results are used to characterize the ergodic rate region of multiple antenna multiple access channels, when each point-to-point propagation channel is modelled according to the Kronecker model. Specifically, an approximation of all rates achieved within the ergodic rate region is derived and an approximation of the linear precoders that achieve the boundary of the rate region as well as an iterative water-filling algorithm to obtain these precoders are provided. An original feature of this work is that the proposed deterministic equivalents are proved valid even for strong correlation patterns at both communication sides. The above results are validated by Monte Carlo simulations.}, number={6}, journal={IEEE Transactions on Information Theory}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Couillet, Romain and Debbah, Mérouane and Silverstein, Jack W.}, year={2011}, month={Jun}, pages={3493–3514} }
 @article{couillet_silverstein_bai_debbah_2011, title={Eigen-Inference for Energy Estimation of Multiple Sources}, volume={57}, ISSN={["1557-9654"]}, DOI={10.1109/tit.2011.2109990}, abstractNote={In this paper, a new method is introduced to blindly estimate the transmit power of multiple signal sources in multiantenna fading channels, when the number of sensing devices and the number of available samples are sufficiently large compared to the number of sources. Recent advances in the field of large dimensional random matrix theory are used that result in a simple and computationally efficient consistent estimator of the power of each source. A criterion to determine the minimum number of sensors and the minimum number of samples required to achieve source separation is then introduced. Simulations are performed that corroborate the theoretical claims and show that the proposed power estimator largely outperforms alternative power inference techniques.}, number={4}, journal={IEEE TRANSACTIONS ON INFORMATION THEORY}, author={Couillet, Romain and Silverstein, Jack W. and Bai, Zhidong and Debbah, Merouane}, year={2011}, month={Apr}, pages={2420–2439} }
 @inbook{silverstein_2011, title={Random Matrix Theory}, ISBN={9783642048975 9783642048982}, url={http://dx.doi.org/10.1007/978-3-642-04898-2_472}, DOI={10.1007/978-3-642-04898-2_472}, booktitle={International Encyclopedia of Statistical Science}, publisher={Springer Berlin Heidelberg}, author={Silverstein, Jack W.}, year={2011}, pages={1168–1170} }
 @inproceedings{couillet_silverstein_debbah_2010, title={Eigen-inference for multi-source power estimation}, ISBN={9781424478927 9781424478903 9781424478910}, url={http://dx.doi.org/10.1109/isit.2010.5513327}, DOI={10.1109/isit.2010.5513327}, abstractNote={This paper introduces a new method to estimate the power transmitted by multiple signal sources, when the number of sensing devices and the available samples are sufficiently large compared to the number of sources. This work makes use of recent advances in the field of random matrix theory that prove more efficient than previous “moment-based” approaches to the problem of multi-source power detection. Simulations are performed which corroborate the theoretical claims.}, booktitle={2010 IEEE International Symposium on Information Theory}, publisher={IEEE}, author={Couillet, Romain and Silverstein, Jack W. and Debbah, Merouane}, year={2010}, month={Jun} }
 @article{nadakuditi_silverstein_2010, title={Fundamental Limit of Sample Generalized Eigenvalue Based Detection of Signals in Noise Using Relatively Few Signal-Bearing and Noise-Only Samples}, volume={4}, ISSN={["1932-4553"]}, DOI={10.1109/jstsp.2009.2038310}, abstractNote={The detection problem in statistical signal processing can be succinctly formulated: given m (possibly) signal bearing, n -dimensional signal-plus-noise snapshot vectors (samples) and N statistically independent n-dimensional noise-only snapshot vectors, can one reliably infer the presence of a signal? This problem arises in the context of applications as diverse as radar, sonar, wireless communications, bioinformatics, and machine learning and is the critical first step in the subsequent signal parameter estimation phase. The signal detection problem can be naturally posed in terms of the sample generalized eigenvalues. The sample generalized eigenvalues correspond to the eigenvalues of the matrix formed by ?whitening? the signal-plus-noise sample covariance matrix with the noise-only sample covariance matrix. In this paper, we prove a fundamental asymptotic limit of sample generalized eigenvalue-based detection of signals in arbitrarily colored noise when there are relatively few signal bearing and noise-only samples. Specifically, we show why when the (eigen) signal-to-noise ratio (SNR) is below a critical value, that is a simple function of n , m, and N, then reliable signal detection, in an asymptotic sense, is not possible. If, however, the eigen-SNR is above this critical value then a simple, new random matrix theory-based algorithm, which we present here, will reliably detect the signal even at SNRs close to the critical value. Numerical simulations highlight the accuracy of our analytical prediction, permit us to extend our heuristic definition of the effective number of identifiable signals in colored noise and display the dramatic improvement in performance relative to the classical estimator by Zhao We discuss implications of our result for the detection of weak and/or closely spaced signals in sensor array processing, abrupt change detection in sensor networks, and clustering methodologies in machine learning.}, number={3}, journal={IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING}, author={Nadakuditi, Raj Rao and Silverstein, Jack W.}, year={2010}, month={Jun}, pages={468–480} }
 @book{bai_silverstein_2010, place={New York}, edition={2}, title={Spectral Analysis of Large Dimensional Random Matrices}, publisher={Springer}, author={Bai, Z.D. and Silverstein, Jack}, year={2010} }
 @inproceedings{couillet_debbah_silverstein_2009, title={Asymptotic capacity of multi-user MIMO communications}, ISBN={9781424449828}, url={http://dx.doi.org/10.1109/itw.2009.5351274}, DOI={10.1109/itw.2009.5351274}, abstractNote={This paper introduces two new formulas to derive explicit capacity expressions of a class of communication schemes, which include single-cell multi-user MIMO and single-user MIMO with multi-cell interference. The extension of a classical theorem from Silverstein allows us to assume a channel Kronecker model between the base stations and the cellular terminals, provided that they all embed a large number of antennas. As an introductory example, we study the single-user MIMO setting with multi-cell interference, in the downlink. We provide new asymptotic capacity formulas when single-user decoding of the incoming data or MMSE decoding are used. Simulations are shown to corroborate the theoretical claims, even when the number of transmit/receive antennas is not very large.}, booktitle={2009 IEEE Information Theory Workshop}, publisher={IEEE}, author={Couillet, Romain and Debbah, Merouane and Silverstein, Jack W.}, year={2009}, month={Oct} }
 @article{paul_silverstein_2009, title={No eigenvalues outside the support of the limiting empirical spectral distribution of a separable covariance matrix}, volume={100}, ISSN={["0047-259X"]}, DOI={10.1016/j.jmva.2008.03.010}, abstractNote={We consider a class of matrices of the form Cn=(1/N)An1/2XnBnXn∗×An1/2, where Xn is an n×N matrix consisting of i.i.d. standardized complex entries, An1/2 is a nonnegative definite square root of the nonnegative definite Hermitian matrix An, and Bn is diagonal with nonnegative diagonal entries. Under the assumption that the distributions of the eigenvalues of An and Bn converge to proper probability distributions as nN→c∈(0,∞), the empirical spectral distribution of Cn converges a.s. to a non-random limit. We show that, under appropriate conditions on the eigenvalues of An and Bn, with probability 1, there will be no eigenvalues in any closed interval outside the support of the limiting distribution, for sufficiently large n. The problem is motivated by applications in spatio-temporal statistics and wireless communications.}, number={1}, journal={JOURNAL OF MULTIVARIATE ANALYSIS}, author={Paul, Debashis and Silverstein, Jack W.}, year={2009}, month={Jan}, pages={37–57} }
 @inproceedings{couillet_debbah_silverstein_2009, title={Rate region of correlated MIMO multiple access channels and broadcast channels}, ISBN={9781424427093}, url={http://dx.doi.org/10.1109/ssp.2009.5278602}, DOI={10.1109/ssp.2009.5278602}, abstractNote={In this paper, the rate region of large multi-antenna multiple access channels and broadcast channels are investigated. The propagation channels between transmitters and receivers are modelled as independent Gaussian with separable variance profiles. It is shown in particular that the large antenna rate regions do not depend on the specific channel realization, but only on the channel transmit and receive covariance matrices. The theoretical results are corroborated by simulations.}, booktitle={2009 IEEE/SP 15th Workshop on Statistical Signal Processing}, publisher={IEEE}, author={Couillet, Romain and Debbah, Merouane and Silverstein, Jack W.}, year={2009}, month={Aug} }
 @inbook{silverstein_2009, title={THE STIELTJES TRANSFORM AND ITS ROLE IN EIGENVALUE BEHAVIOR OF LARGE DIMENSIONAL RANDOM MATRICES}, ISBN={9789814273114 9789814273121}, ISSN={1793-0758}, url={http://dx.doi.org/10.1142/9789814273121_0001}, DOI={10.1142/9789814273121_0001}, booktitle={Random Matrix Theory and Its Applications}, publisher={WORLD SCIENTIFIC}, author={Silverstein, Jack W.}, year={2009}, month={Jul}, pages={1–25} }
 @article{dozier_silverstein_2007, title={Analysis of the limiting spectral distribution of large dimensional information-plus-noise type matrices}, volume={98}, ISSN={["0047-259X"]}, DOI={10.1016/j.jmva.2006.12.005}, abstractNote={A derivation of results on the analytic behavior of the limiting spectral distribution of sample covariance matrices of the “information-plus-noise” type, as studied in Dozier and Silverstein [On the empirical distribution of eigenvalues of large dimensional information-plus-noise type matrices, 2004, submitted for publication], is presented. It is shown that, away from zero, the limiting distribution possesses a continuous density. The density is analytic where it is positive and, for the most relevant cases of a in the boundary of its support, exhibits behavior closely resembling that of |x-a| for x near a. A procedure to determine its support is also analyzed.}, number={6}, journal={JOURNAL OF MULTIVARIATE ANALYSIS}, author={Dozier, R. Brent and Silverstein, Jack W.}, year={2007}, month={Jul}, pages={1099–1122} }
 @inproceedings{nadakuditi_silverstein_2007, title={Fundamental Limit of Sample Eigenvalue based Detection of Signals in Colored Noise using Relatively Few Samples}, ISBN={9781424421091 9781424421107}, ISSN={1058-6393}, url={http://dx.doi.org/10.1109/acssc.2007.4487301}, DOI={10.1109/acssc.2007.4487301}, abstractNote={Sample eigenvalue based estimators are often used for estimating the number of high-dimensional signals in colored noise when an independent estimate of the noise covariance matrix is available. We highlight a fundamental asymptotic limit of sample eigenvalue based detection that brings into sharp focus why in the large system, relatively large sample size limit, underestimation of the model order may be unavoidable for weak/closely spaced signals. We discuss the implication of these results for the detection of two weak, closely spaced signals.}, booktitle={2007 Conference Record of the Forty-First Asilomar Conference on Signals, Systems and Computers}, publisher={IEEE}, author={Nadakuditi, Raj Rao and Silverstein, Jack W.}, year={2007}, month={Nov} }
 @article{dozier_silverstein_2007, title={On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices}, volume={98}, ISSN={["0047-259X"]}, DOI={10.1016/j.jmva.2006.09.006}, abstractNote={Let Xn be n×N containing i.i.d. complex entries and unit variance (sum of variances of real and imaginary parts equals 1), σ>0 constant, and Rn an n×N random matrix independent of Xn. Assume, almost surely, as n→∞, the empirical distribution function (e.d.f.) of the eigenvalues of 1NRnRn* converges in distribution to a nonrandom probability distribution function (p.d.f.), and the ratio nN tends to a positive number. Then it is shown that, almost surely, the e.d.f. of the eigenvalues of 1N(Rn+σXn)(Rn+σXn)* converges in distribution. The limit is nonrandom and is characterized in terms of its Stieltjes transform, which satisfies a certain equation.}, number={4}, journal={JOURNAL OF MULTIVARIATE ANALYSIS}, author={Dozier, R. Brent and Silverstein, Jack W.}, year={2007}, month={Apr}, pages={678–694} }
 @article{bai_silverstein_2007, title={On the signal-to-interference ratio of CDMA systems in wireless communications}, volume={17}, ISSN={["1050-5164"]}, DOI={10.1214/105051606000000637}, abstractNote={Let $\{s_{ij}:i,j=1,2,...\}$ consist of i.i.d. random variables in $\mathbb{C}$ with $\mathsf{E}s_{11}=0$, $\mathsf{E}|s_{11}|^2=1$. For each positive integer $N$, let $\mathbf{s}_k={\mathbf{s}}_k(N)=(s_{1k},s_{2k},...,s_{Nk})^T$, $1\leq k\leq K$, with $K=K(N)$ and $K/N\to c>0$ as $N\to\infty$. Assume for fixed positive integer $L$, for each $N$ and $k\leq K$, ${\bolds\alpha}_k=(\alpha_k(1),...,\alpha_k(L))^T$ is random, independent of the $s_{ij}$, and the empirical distribution of $(\alpha_1,...,\alpha_K)$, with probability one converging weakly to a probability distribution $H$ on $\mathbb{C}^L$. Let ${\bolds\beta }_k={\bolds\beta}_k(N)=(\alpha_k(1)\mathbf{s}_k^T,...,\alpha_k(L)\m athbf{s}_k^T)^T$ and set $C=C(N)=(1/N)\sum_{k=2}^K{\bolds \beta}_k{\bolds \beta}_k^*$. Let $\sigma^2>0$ be arbitrary. Then define $SIR_1=(1/N){\bolds\beta}^*_1(C+\sigma^2I)^{-1}{\bolds\beta}_1$, which represents the best signal-to-interference ratio for user 1 with respect to the other $K-1$ users in a direct-sequence code-division multiple-access system in wireless communications. In this paper it is proven that, with probability 1, $SIR_1$ tends, as $N\to\infty$, to the limit $\sum_{\ell,\ell'=1}^L\bar{\alpha}_1(\ell) alpha_1(\ell')a_{\ell,\ell'},$ where $A=(a_{\ell,\ell'})$ is nonrandom, Hermitian positive definite, and is the unique matrix of such type satisfying $A=\bigl(c \mathsf{E}\frac{{\bolds\alpha}{\bolds \alpha}^*}{1+{\bolds\alpha}^*A{\bolds\alpha}}+\sigma^2I_L\bigr)^{-1}$, where ${\bolds\alpha}\in \mathbb{C}^L$ has distribution $H$. The result generalizes those previously derived under more restricted assumptions.}, number={1}, journal={ANNALS OF APPLIED PROBABILITY}, author={Bai, Z. D. and Silverstein, Jack W.}, year={2007}, month={Feb}, pages={81–101} }
 @article{baik_silverstein_2006, title={Eigenvalues of large sample covariance matrices of spiked population models}, volume={97}, ISSN={["0047-259X"]}, DOI={10.1016/j.jmva.2005.08.003}, abstractNote={We consider a spiked population model, proposed by Johnstone, in which all the population eigenvalues are one except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the non-unit population ones when both sample size and population size become large. This paper completely determines the almost sure limits of the sample eigenvalues in a spiked model for a general class of samples.}, number={6}, journal={JOURNAL OF MULTIVARIATE ANALYSIS}, author={Baik, Jinho and Silverstein, Jack W.}, year={2006}, month={Jul}, pages={1382–1408} }
 @article{rider_silverstein_2006, title={Gaussian fluctuations for non-Hermitian random matrix ensembles}, volume={34}, ISSN={["0091-1798"]}, DOI={10.1214/009117906000000403}, abstractNote={Consider an ensemble of N×N non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry distributions also possess bounded densities and finite (4+ɛ) moments, then Z. D. Bai [Ann. Probab. 25 (1997) 494–529] has shown the ensemble to satisfy the circular law: after scaling by a factor of $1/\sqrt{N}$ and letting N→∞, the empirical measure of the eigenvalues converges weakly to the uniform measure on the unit disk in the complex plane. In this note, we investigate fluctuations from the circular law in a more restrictive class of non-Hermitian matrices for which higher moments of the entries obey a growth condition. The main result is a central limit theorem for linear statistics of type XN(f)=∑k=1Nf(λk) where λ1, λ2, …, λN denote the ensemble eigenvalues and the test function f is analytic on an appropriate domain. The proof is inspired by Bai and Silverstein [Ann. Probab. 32 (2004) 533–605], where the analogous result for random sample covariance matrices is established.}, number={6}, journal={ANNALS OF PROBABILITY}, author={Rider, B. and Silverstein, Jack W.}, year={2006}, month={Nov}, pages={2118–2143} }
 @book{bai_silverstein_2006, place={New York / Beijing}, edition={1}, title={Spectral Analysis of Large Dimensional Random Matrices}, publisher={Springer / Science Press}, author={Bai, Z.D. and Silverstein, Jack}, year={2006} }
 @inproceedings{silverstein_tulino_2006, title={Theory of Large Dimensional Random Matrices for Engineers}, ISBN={0780397800 0780397797}, url={http://dx.doi.org/10.1109/isssta.2006.311814}, DOI={10.1109/isssta.2006.311814}, abstractNote={In the last few years, the asymptotic distribution of the singular values of certain random matrices has emerged as a key tool in the analysis and design of wireless communication channels. These channels are characterized by random matrices that admit various statistical descriptions depending on the actual application. The goal of this paper is the investigation and application of random matrix theory with particular emphasis on the asymptotic theorems on the distribution of the squared singular values under various assumption on the joint distribution of the random matrix entries}, booktitle={2006 IEEE Ninth International Symposium on Spread Spectrum Techniques and Applications}, publisher={IEEE}, author={Silverstein, Jack and Tulino, Antonia}, year={2006}, month={Aug} }
 @article{bai_silverstein_2004, title={CLT for linear spectral statistics of large-dimensional sample covariance matrices}, volume={32}, number={1A}, journal={Annals of Probability}, author={Bai, Z. D. and Silverstein, J. W.}, year={2004}, pages={553–605} }
 @article{bai_silverstein_1999, title={Exact separation of eigenvalues of large dimensional sample covariance matrices}, volume={27}, number={3}, journal={Annals of Probability}, author={Bai, Z. D. and Silverstein, J. W.}, year={1999}, pages={1536–1555} }
 @article{silverstein_1999, title={Methodologies in spectral analysis of large dimensional random matrices, a review - Comment: Complements and new developments}, volume={9}, number={3}, journal={Statistica Sinica}, author={Silverstein, J. W.}, year={1999}, pages={667–671} }
 @article{bai_silverstein_1998, title={No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices}, volume={26}, DOI={10.1214/aop/1022855421}, abstractNote={Let B n = (1/N)T n 1/2 X n X n *Tn 1/2 , where X n is n x N with i.i.d. complex standardized entries having finite fourth moment and T n 1/2 is a Hermitian square root of the nonnegative definite Hermitian matrix T n . It is known that, as n → ∞, if n/N converges to a positive number and the empirical distribution of the eigenvalues of T n converges to a proper probability distribution, then the empirical distribution of the eigenvalues of B n converges a.s. to a nonrandom limit. In this paper we prove that, under certain conditions on the eigenvalues of T n , for any closed interval outside the support of the limit, with probability 1 there will be no eigenvalues in this interval for all n sufficiently large.}, number={1}, journal={Annals of Probability}, author={Bai, Z. D. and Silverstein, Jack W.}, year={1998}, pages={316–345} }
 @article{silverstein_choi_1995, title={Analysis of the Limiting Spectral Distribution of Large Dimensional Random Matrices}, volume={54}, ISSN={0047-259X}, url={http://dx.doi.org/10.1006/jmva.1995.1058}, DOI={10.1006/jmva.1995.1058}, abstractNote={Results on the analytic behavior of the limiting spectral distribution of matrices of sample covariance type, studied in Marcenko and Pastur [2] and Yin [8], are derived. Through an equation defining its Stieltjes transform, it is shown that the limiting distribution has a continuous derivative away from zero, the derivative being analytic wherever it is positive, and resembles [formula] for most cases of x0 in the boundary of its support. A complete analysis of a way to determine its support, originally outlined in Marcenko and Pastur [2], is also presented.}, number={2}, journal={Journal of Multivariate Analysis}, publisher={Elsevier BV}, author={Silverstein, J.W. and Choi, S.I.}, year={1995}, month={Aug}, pages={295–309} }
 @article{silverstein_bai_1995, title={On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices}, volume={54}, ISSN={0047-259X}, url={http://dx.doi.org/10.1006/jmva.1995.1051}, DOI={10.1006/jmva.1995.1051}, abstractNote={A stronger result on the limiting distribution of the eigenvalues of random Hermitian matrices of the form A + XTX*, originally studied in Marcenko and Pastur, is presented. Here, X(N - n), T(n - n), and A(N - N) are independent, with X containing i.i.d. entries having finite second moments, T is diagonal with real (diagonal) entries, A is Hermitian, and n/N --> c > 0 as N --> [infinity]. Under additional assumptions on the eigenvalues of A and T, almost sure convergence of the empirical distribution function of the eigenvalues of A + XTX* is proven with the aid of Stieltjes transforms, taking a more direct approach than previous methods.}, number={2}, journal={Journal of Multivariate Analysis}, publisher={Elsevier BV}, author={Silverstein, J.W. and Bai, Z.D.}, year={1995}, month={Aug}, pages={175–192} }
 @article{silverstein_1995, title={Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices}, volume={55}, ISSN={0047-259X}, url={http://dx.doi.org/10.1006/jmva.1995.1083}, DOI={10.1006/jmva.1995.1083}, abstractNote={Let X be n - N containing i.i.d. complex entries with E X11 - EX112 = 1, and T an n - n random Hermitian nonnegative definite, independent of X. Assume, almost surely, as n --> [infinity], the empirical distribution function (e.d.f.) of the eigenvalues of T converges in distribution, and the ratio n/N tends to a positive number. Then it is shown that, almost surely, the e.d.f. of the eigenvalues of (1/N) XX*T converges in distribution. The limit is nonrandom and is characterized in terms of its Stieltjes transform, which satisfies a certain equation.}, number={2}, journal={Journal of Multivariate Analysis}, publisher={Elsevier BV}, author={Silverstein, J.W.}, year={1995}, month={Nov}, pages={331–339} }
 @article{silverstein_1994, title={The spectral radii and norms of large dimensional non-central random matrices}, volume={10}, ISSN={0882-0287}, url={http://dx.doi.org/10.1080/15326349408807308}, DOI={10.1080/15326349408807308}, abstractNote={Consider a matrix made up of i.i.d. random variables with positive mean and nite fourth moment. Results are given on its spectral norm and (if}, number={3}, journal={Communications in Statistics. Stochastic Models}, publisher={Informa UK Limited}, author={Silverstein, Jack W.}, year={1994}, month={Jan}, pages={525–532} }
 @inproceedings{silverstein_combettes_1992, place={Hoboken, New Jersey}, title={Large dimensional random matrix theory for signal detection and estimation in array processing}, ISBN={0780305086}, url={http://dx.doi.org/10.1109/ssap.1992.246796}, DOI={10.1109/ssap.1992.246796}, abstractNote={This paper brings into play elements of the spectral theory of such matrices and demonstrates their relevance to source detection and bearing estimation in problems with sizable arrays. These results are applied to the sample spatial covariance matrix, R, of the sensed data. It is seen that detection can be achieved with a sample size considerably less than that required by conventional approaches. It is argued that more accurate estimates of direction of arrival can be obtained by constraining R to be consistent with various a priori constraints including those arising from large dimensional random matrix theory. A set theoretic formalism is used for this feasibility problem. Unsolved issues are discussed.<<ETX>>}, booktitle={[1992] IEEE Sixth SP Workshop on Statistical Signal and Array Processing}, publisher={IEEE}, author={Silverstein, J.W. and Combettes, P.L.}, year={1992}, pages={276–279} }
 @article{silverstein_combettes_1992, title={Signal detection via spectral theory of large dimensional random matrices}, volume={40}, ISSN={1053-587X}, url={http://dx.doi.org/10.1109/78.149981}, DOI={10.1109/78.149981}, abstractNote={Results on the spectral behavior of random matrices as the dimension increases are applied to the problem of detecting the number of sources impinging on an array of sensors. A common strategy to solve this problem is to estimate the multiplicity of the smallest eigenvalue of the spatial covariance matrix R of the sensed data. Existing approaches rely on the closeness of the noise eigenvalues of sample covariance matrix to each other and, therefore, the sample size has to be quite large when the number of sources is large in order to obtain a good estimate. The theoretical analysis presented focuses on the splitting of the spectrum of sample covariance matrix into noise and signal eigenvalues. It is shown that when the number of sensors is large the number of signals can be estimated with a sample size considerably less than that required by previous approaches. >}, number={8}, journal={IEEE Transactions on Signal Processing}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Silverstein, J.W. and Combettes, P.L.}, year={1992}, pages={2100–2105} }
 @book{silverstein_combettes_1990, title={Spectral theory of large dimensional random matrices applied to signal detection}, author={Silverstein, Jack W. and Combettes, Patrick L.}, year={1990} }
 @article{silverstein_1990, title={Weak Convergence of Random Functions Defined by The Eigenvectors of Sample Covariance Matrices}, volume={18}, ISSN={0091-1798}, url={http://dx.doi.org/10.1214/aop/1176990741}, DOI={10.1214/aop/1176990741}, abstractNote={Let {v ij }, i, j = 1, 2,..., be i.i.d. symmetric random variables with E(v 4 11 ) 0 as n → ∞. Denote by O n Λ n O T n the spectral decomposition of M n . Define X ∈ D[0,1] by X n (t) = √n/2Σ [nt] i=1 (y 2 i ― 1/n), where (y 1 ,y 2 ,..., y n ) T =O T (±1/√n, ±1/√n,..., ± 1/√n) T . It is shown that X n → D W 0 as n → ∞, where W 0 is a Brownian bridge. This result sheds some light on the problem of describing the behavior of the eigenvectors of M n for n large and for general v 11}, number={3}, journal={The Annals of Probability}, publisher={Institute of Mathematical Statistics}, author={Silverstein, Jack W.}, year={1990}, month={Jul}, pages={1174–1194} }
 @article{silverstein_1989, title={On the eigenvectors of large dimensional sample covariance matrices}, volume={30}, ISSN={0047-259X}, url={http://dx.doi.org/10.1016/0047-259x(89)90084-5}, DOI={10.1016/0047-259x(89)90084-5}, abstractNote={Let {vij}, i, j = 1,2, …, be i.i.d. random variables, and for each n let Mn = (1s)VnVnT, where Vn = (vij), i = 1, 2, …, n, j = 1, 2, …, s = s(n), and ns → y > 0 as n → ∞. Necessary and sufficient conditions are given to establish the convergence in distribution of certain random variables defined by Mn. When E(v114) < ∞ these variables play an important role toward understanding the behavior of the eigenvectors of this class of sample covariance matrices for n large.}, number={1}, journal={Journal of Multivariate Analysis}, publisher={Elsevier BV}, author={Silverstein, Jack W.}, year={1989}, month={Jul}, pages={1–16} }
 @article{silverstein_1989, title={On the weak limit of the largest eigenvalue of a large dimensional sample covariance matrix}, volume={30}, ISSN={0047-259X}, url={http://dx.doi.org/10.1016/0047-259x(89)90042-0}, DOI={10.1016/0047-259x(89)90042-0}, abstractNote={Let {wij}, i, j = 1, 2, …, be i.i.d. random variables and for each n let Mn = (1n) WnWnT, where Wn = (wij), i = 1, 2, …, p; j = 1, 2, …, n; p = p(n), and pn → y > 0 as n → ∞. The weak behavior of the largest eigenvalue of Mn is studied. The primary aim of the paper is to show that the largest eigenvalue converges in probability to a nonrandom quantity if and only if E(w11) = 0 and n4P(|ω11| ≥ n) = o(1), the limit being (1 + √y)2 E(w112).}, number={2}, journal={Journal of Multivariate Analysis}, publisher={Elsevier BV}, author={Silverstein, Jack W}, year={1989}, month={Aug}, pages={307–311} }
 @article{bai_silverstein_yin_1988, title={A note on the largest eigenvalue of a large dimensional sample covariance matrix}, volume={26}, ISSN={0047-259X}, url={http://dx.doi.org/10.1016/0047-259x(88)90078-4}, DOI={10.1016/0047-259x(88)90078-4}, abstractNote={Let {vij; i, j = 1, 2, …} be a family of i.i.d. random variables with E(v114) = ∞. For positive integers p, n with p = p(n) and pn → y > 0 as n → ∞, let Mn = (1n) Vn VnT , where Vn = (vij)1 ≤ i ≤ p, 1 ≤ j ≤ n, and let λmax(n) denote the largest eigenvalue of Mn. It is shown that limnλmax(n) = ∞ a.s. This result verifies the boundedness of E(v114) to be the weakest condition known to assure the almost sure convergence of λmax(n) for a class of sample covariance matrices.}, number={2}, journal={Journal of Multivariate Analysis}, publisher={Elsevier BV}, author={Bai, Z.D and Silverstein, Jack W and Yin, Y.Q}, year={1988}, month={Aug}, pages={166–168} }
 @article{silverstein_1986, place={Providence}, title={Eigenvalues and eigenvectors of large-dimensional sample covariance matrices}, volume={50}, ISBN={9780821850442 9780821876350}, ISSN={1098-3627 0271-4132}, url={http://dx.doi.org/10.1090/conm/050/841089}, DOI={10.1090/conm/050/841089}, journal={Random Matrices and Their Applications}, publisher={American Mathematical Society}, author={Silverstein, Jack W.}, year={1986}, pages={153–159} }
 @article{silverstein_1985, title={The Limiting Eigenvalue Distribution of a Multivariate F Matrix}, volume={16}, ISSN={0036-1410 1095-7154}, url={http://dx.doi.org/10.1137/0516047}, DOI={10.1137/0516047}, abstractNote={Let $X_{ij} ,Y_{ij} i,j = 1,2, \cdots $, be i.i.d. $N(0,1)$ random variables and for positive integers $p,m,n $, let $\bar X_p = (X_{ij} ) i = 1,2, \cdots ,p; j = 1,2, \cdots ,m$, and $\bar Y_p = (Y_{ij} ) i = 1,2, \cdots ,p; j = 1,2, \cdots ,n$. Suppose further that ${p / {m \to y > 0}}$ and ${p / {n \to y' \in (0,\frac{1}{2})}}$ as $p \to \infty $ . In [5], [6] it is shown that the empirical distribution function of the eigenvalues of $({1 / m}\bar X_p \bar X_p^T )({1 / n}\bar Y_p \bar Y_p^T )^{ - 1}$ converges i.p. as $p \to \infty $ to a nonrandom d.f.In the present paper the limiting d.f. is derived.}, number={3}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Silverstein, Jack W.}, year={1985}, month={May}, pages={641–646} }
 @article{silverstein_1985, title={The Smallest Eigenvalue of a Large Dimensional Wishart Matrix}, volume={13}, ISSN={0091-1798}, url={http://dx.doi.org/10.1214/aop/1176992819}, DOI={10.1214/aop/1176992819}, abstractNote={For each s : 1,2... let n: r?(s) be a posit ive integer such that n/s --+ y > 0 as s --+ cc. Let V" be an n X s matrix whose entries are i.i.d. N(0, 1) random variables and let M": (I/s)V"V:.The random matrix V,V: is commonly referred to as the Wishart matrix W(1,, s). It is well known [Mar0enko and Pastur (1967), Wachter (1978)] that the empirical distribution function 4 of the eigenvalues of M, t.F"(r) = (I/n) x (number of eigenvalues of M " < *)l converges almost r.rr"ly ur r -+ oo to a nonrandom probability distribution function { having a density with positive support on [(1 tF)',(l + ,F>t], and when yi r, { yields additional mass on {0}. It is also known fGeman (1980)] that the maximum eigenvalue I(** of M" converges a.s. to (1 + ,F)'as s -+ oo. [The statement of thi. r"rult in Geman (1980) has all the M, constructed from one doubly infinite array of i.i.d. random variables. However, it is obvious from the proof that no relation on the entries of V" for different s is needed.] These results are established under assumptions more general on the entries of % than Gaussian distributed, involving conditions on the moments of these random variables. The present paper will prove the following}, number={4}, journal={The Annals of Probability}, publisher={Institute of Mathematical Statistics}, author={Silverstein, Jack W.}, year={1985}, month={Nov}, pages={1364–1368} }
 @article{silverstein_1984, title={Comments on a result of Yin, Bai, and Krishnaiah for large dimensional multivariate F matrices}, volume={15}, ISSN={0047-259X}, url={http://dx.doi.org/10.1016/0047-259x(84)90059-9}, DOI={10.1016/0047-259x(84)90059-9}, abstractNote={A theorem in Yin, Bai, and Krishnaiah ( J. Multivariate Anal. 13 (1983), 508–516) shows that the smallest eigenvalue of a class of large dimensional sample covariance matrices stays almost surely bounded away from zero. The theorem assumes a certain restriction on the class of matrices. With slight modifications of the proof in op cit, it is shown here that the theorem is true for all relevant matrices.}, number={3}, journal={Journal of Multivariate Analysis}, publisher={Elsevier BV}, author={Silverstein, Jack W}, year={1984}, month={Dec}, pages={408–409} }
 @book{yin_bai_krishnaiah_1984, title={On Limit of the Largest Eigenvalue of the Large Dimensional Sample Covariance Matrix.}, url={http://dx.doi.org/10.21236/ada150589}, DOI={10.21236/ada150589}, abstractNote={Abstract : The authors showed that the largest eigenvalue of the sample covariance matrix tends to a limit under certain conditions when both the number of variables and the sample size tend to infinity. The above result is proved under the mild restriction that the fourth moment of the elements of the sample sums of squares and cross products (SP) matrix exist. Key words include: Largest eigenvalue, Sample covariance matrix, Large dimensional random matrices, Limit.}, institution={Defense Technical Information Center}, author={Yin, Y. Q. and Bai, Z. D. and Krishnaiah, P. R.}, year={1984}, month={Oct} }
 @article{silverstein_1984, title={Some limit theorems on the eigenvectors of large dimensional sample covariance matrices}, volume={15}, ISSN={0047-259X}, url={http://dx.doi.org/10.1016/0047-259x(84)90054-x}, DOI={10.1016/0047-259x(84)90054-x}, abstractNote={Let {vij} i,j = 1, 2,…, be i.i.d. standardized random variables. For each n, let Vn = (vij) i = 1, 2,…, n; j = 1, 2,…, s = s(n), where (ns) → y > 0 as n → ∞, and let Mn = (1s)VnVnT. Previous results [7, 8] have shown the eigenvectors of Mn to display behavior, for n large, similar to those of the corresponding Wishart matrix. A certain stochastic process Xn on [0, 1], constructed from the eigenvectors of Mn, is known to converge weakly, as n → ∞, on D[0, 1] to Brownian bridge when v11 is N(0, 1), but it is not known whether this property holds for any other distribution. The present paper provides evidence that this property may hold in the non-Wishart case in the form of limit theorems on the convergence in distribution of random variables constructed from integrating analytic function w.r.t. Xn(Fn(x)), where Fn is the empirical distribution function of the eigenvalues of Mn. The theorems assume certain conditions on the moments of v11 including E(v114) = 3, the latter being necessary for the theorems to hold.}, number={3}, journal={Journal of Multivariate Analysis}, publisher={Elsevier BV}, author={Silverstein, Jack W}, year={1984}, month={Dec}, pages={295–324} }
 @article{campbell_silverstein_1981, title={A nonlinear system with singular vector field near equilibria}, volume={12}, ISSN={0003-6811 1563-504X}, url={http://dx.doi.org/10.1080/00036818108839348}, DOI={10.1080/00036818108839348}, abstractNote={The system Nω=(N-α)ω+y, N= bN+aωωT, N(t)∊Rm×m, ω(t)∊Rm which originally arose from a model for the pathological behavior of neural networks, is studied. Similar equations can arise in a variety of applications. It is shown that if N(0) is positive definite, then solutions exist for all time. Equilibrium points are determined. N is found to be singular at the equilibrium points, making the analysis of the asymptotic properties of the system non-trivial. The asymptotic behavior when y = 0 is completely described. Some results are proven on the asymptotic behavior of N and ω when y≠0}, number={1}, journal={Applicable Analysis}, publisher={Informa UK Limited}, author={Campbell, Stephen L. and Silverstein, Jack W.}, year={1981}, month={Jan}, pages={57–71} }
 @article{silverstein_1981, title={Describing the Behavior of Eigenvectors of Random Matrices Using Sequences of Measures on Orthogonal Groups}, volume={12}, ISSN={0036-1410 1095-7154}, url={http://dx.doi.org/10.1137/0512025}, DOI={10.1137/0512025}, abstractNote={A conjecture has previously been made on the chaotic behavior of the eigenvectors of a class of n-dimensional random matrices, where n is very large [J. Silverstein, SIAM J. Appl. Math., 37 (1979), pp. 235–245]. Evidence supporting the conjecture has been given in the form of two limit theorems, as $n \to \infty $, relating the random matrices to matrices formed from the Haar measure, $h_n $, on the orthogonal group $\mathcal{O}_n $.The present paper considers a reformulation of the conjecture in terms of sequences of the form ${ {\mu _n } }$, where for each n, $\mu _n $ is a Borel probability measure on $\mathcal{O}_n $. A characterization of $\mu _n $ being “close” to $h_n $ for n large is developed. It is suggested that before a definition of what it means for $\{ {\mu _n } \}$ to be asymptotic Haar is decided, properties $\{ {h_n } \}$ possess should first be proposed as possible necessary conditions. The limit theorems are converted into properties on $\{ {\mu _n } \}$. It is shown (Theorem 1) that o...}, number={2}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Silverstein, Jack W.}, year={1981}, month={Mar}, pages={274–281} }
 @article{silverstein_1979, title={On the Randomness of Eigenvectors Generated from Networks with Random Topologies}, volume={37}, ISSN={0036-1399 1095-712X}, url={http://dx.doi.org/10.1137/0137014}, DOI={10.1137/0137014}, abstractNote={A model for the generation of neural connections at birth led to the study of W, a random, symmetric, nonnegative definite linear operator defined on a finite, but very large, dimensional Euclidean space [1]. A limit law, as the dimension increases, on the eigenvalue spectrum of W was proven, implying that realizations of W (being identified with organisms in a species) appear totally different on the microscopic level and yet have almost identical spectral densities.The present paper considers the eigenvectors of W. Evidence is given to support the conjecture that, contrary to the deterministic aspect of the eigenvalues, the eigenvectors behave in a completely chaotic manner, which is described in terms of the normalized uniform (Haar) measure on the group of orthogonal transformations on a finite dimensional space. The validity of the conjecture would imply a tabula rasa property on the ensemble (“species”) of all realizations of W.}, number={2}, journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Silverstein, Jack W.}, year={1979}, month={Oct}, pages={235–245} }
 @book{silverstein_1978, place={Providence, Rhode Island}, title={On word classification using a neural network}, volume={2}, number={4343}, institution={Division of Applied Mathematics, Brown University}, author={Silverstein, Jack}, year={1978} }
 @book{dalenius_silverstein_1978, place={Stockholm, Sweden}, title={Public key cryptosystems: an elementary overview}, number={2929}, journal={Confidentiality in Surveys}, institution={Department of Statistics, University of Stockholm}, author={Dalenius, Tore and Silverstein, Jack}, year={1978} }
 @article{anderson_silverstein_1978, title={Reply to Grossberg.}, volume={85}, ISSN={1939-1471 0033-295X}, url={http://dx.doi.org/10.1037/0033-295x.85.6.597}, DOI={10.1037/0033-295x.85.6.597}, number={6}, journal={Psychological Review}, publisher={American Psychological Association (APA)}, author={Anderson, James A. and Silverstein, Jack W.}, year={1978}, pages={597–603} }
 @article{anderson_silverstein_ritz_jones_1977, title={Distinctive features, categorical perception, and probability learning: Some applications of a neural model.}, volume={84}, ISSN={1939-1471 0033-295X}, url={http://dx.doi.org/10.1037/0033-295x.84.5.413}, DOI={10.1037/0033-295x.84.5.413}, abstractNote={A previously proposed model for memory based on neurophysiolo gical considerations is reviewed. We assume that (a) nervous system activity is usefully represented as the set of simultaneous individual neuron activities in a group of neurons; (b) different memory traces make use of the same synapses; and (c) synapses associate two patterns of neural activity by incrementing synaptic connectivity proportionally to the product of pre- and postsynaptic activity, forming a matrix of synaptic connectivities. We extend this model by (a) introducing positive feedback of a set of neurons onto itself and (b) allowing the individual neurons to saturate. A hybrid model, partly analog and partly binary, arises. The system has certain characteristics reminiscent of analysis by distinctive features. Next, we apply the model to "categorical perception." Finally, we discuss probability learning. The model can predict overshooting, recency data, and probabilities occurring in systems with more than two events with reasonably good accuracy.}, number={5}, journal={Psychological Review}, publisher={American Psychological Association (APA)}, author={Anderson, James A. and Silverstein, Jack W. and Ritz, Stephen A. and Jones, Randall S.}, year={1977}, month={Sep}, pages={413–451} }
 @article{grenander_silverstein_1977, title={Spectral Analysis of Networks with Random Topologies}, volume={32}, ISSN={0036-1399 1095-712X}, url={http://dx.doi.org/10.1137/0132041}, DOI={10.1137/0132041}, abstractNote={A class of neural models is introduced in which the topology of the neural network has been generated by a controlled probability model. It is shown that the resulting linear operator has a spectral measure that converges in probability to a universal one when the size of the net tends to infinity: a law of large numbers for the spectra of such operators. The analytical treatment is accompanied by omputational experiments.}, number={2}, journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Grenander, Ulf and Silverstein, Jack W.}, year={1977}, month={Mar}, pages={499–519} }
 @book{silverstein_1977, place={Providence, Rhode Island}, title={Stability analysis and correction of pathological behavior of neural networks}, number={4848}, institution={Division of Applied Mathematics, Brown University}, author={Silverstein, Jack}, year={1977} }
 @inproceedings{anderson_silverstein_ritz_1977, place={Hartford, Conn}, title={Vowel pre-processing with a neurally based model}, url={http://dx.doi.org/10.1109/icassp.1977.1170257}, DOI={10.1109/icassp.1977.1170257}, abstractNote={We assume that (1) nervous system activity is most usefully represented as the set of simultaneous individual neuron activities in a group of neurons, (2) different memory traces make use of the same synapses and (3) synapses associate two patterns of neural activity by incrementing synaptic connectivity proportional to the product of pre- and post-synaptic activity (a Hebbian rule). We extend this model by (1) introducting positive feedback of a set of neurons onto itself and (2) allowing the individual neurons to saturate. Positive feedback forces the pattern of neural activity into stable corners of a high dimensionality hypercube. The model has behavior reminiscent of 'categorical perception' in that large regions of initial neural activity will end in the same corner. We wish to demonstrate that this model can serve as an efficient pre-processer, which takes a noisy stimulus, a spoken vowel, and puts it in a noise free standard form. As a test, we used acoustic representations of nine spoken Dutch vowels. We apply the model, and show that after several thousand learning trials, input vowels, initially close together, are associated with separate stable corners.}, booktitle={ICASSP '77. IEEE International Conference on Acoustics, Speech, and Signal Processing}, publisher={Institute of Electrical and Electronics Engineers}, author={Anderson, J. and Silverstein, J. and Ritz, S.}, year={1977} }
 @article{silverstein_1976, title={Asymptotics applied to a neural network}, volume={22}, ISSN={0340-1200 1432-0770}, url={http://dx.doi.org/10.1007/bf00320132}, DOI={10.1007/bf00320132}, number={2}, journal={Biological Cybernetics}, publisher={Springer Science and Business Media LLC}, author={Silverstein, Jack W.}, year={1976}, month={Mar}, pages={73–84} }
 @article{silverstein_anderson_ritz_1976, title={Preprocessing with a simple neural model}, volume={2}, number={618}, journal={Neuroscience Abstracts}, author={Silverstein, Jack and Anderson, James and Ritz, Stephen}, year={1976}, pages={438} }
 @book{combettes_silverstein, title={Comments on ‘A Unitary Transformation Method for Angle-of-Arrival Estimation’}, author={Combettes, Patrick L. and Silverstein, Jack} }
 @book{combettes_silverstein, title={Comments on ‘A Unitary Transformation Method for Angle-of-Arrival Estimation’}, author={Combettes, Patrick L. and Silverstein, Jack} }
 @book{combettes_silverstein, title={On the rectification of covariance matrices}, author={Combettes, Patrick L. and Silverstein, Jack} }
 @book{combettes_silverstein, title={On the rectification of covariance matrices}, author={Combettes, Patrick L. and Silverstein, Jack} }