Thesis (MSc (Physics))--University of Stellenbosch, 2005. / It was Wigner that in the 1950’s first introduced the idea of modelling physical reality with an ensemble of random matrices while studying the energy levels of heavy atomic nuclei. Since then, the field of Random Matrix Theory has grown tremendously, with applications ranging from fluctuations on the economic markets to M-theory. It is the purpose of this thesis to discuss the basic concepts of Random Matrix Theory, using the ensembles of random matrices originally introduced by Wigner, the Gaussian ensembles, as a starting point. As Random Matrix Theory is classically concerned with the statistical properties of levels sequences, we start with a brief introduction to the statistical analysis of a level sequence before getting to the introduction of the Gaussian ensembles. With the ensembles defined, we move on to the statistical properties that they predict. In the light of these predictions, a few of the classical applications of Random Matrix Theory are discussed, and as an example of some of the important concepts, the Anderson model of localization is investigated in some detail.

In this thesis we will brush through fundamental multivariate statistical theory and then present MIMO-systems briefly in order to later calculate the channel capacity of a MIMO system. After theory has been presented we will then look at different properties of the channel capacity and then investigate a supposed MIMO system dataset and use standardized methods to verify it’s model. The properties of the limiting channel capacity uses the Marčenko-Pastur law. Therefore we will present some fundamental theorems and definitions of limiting spectral distribution of Wishart and Wigner matrices, and some fundamental properties they hold.

This thesis is about high–dimensional problems considered under the so{called Kolmogorov condition. Hence, we consider research questions related to random matrices with p rows (corresponding to the parameters) and n columns (corresponding to the sample size), where p &gt; n, assuming that the ratio <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%5Cfrac%7Bp%7D%7Bn%7D" /> converges when the number of parameters and the sample size increase. We focus on the eigenvalue distribution of the considered matrices, since it is a well–known information–carrying object. The spectral distribution with compact support is fully characterized by its moments, i.e., by the normalized expectation of the trace of powers of the matrices. Moreover, such an expectation can be seen as a free moment in the non–commutative space of random matrices of size p x p equipped with the functional <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20%5Cfrac%7B1%7D%7Bp%7DE%5BTr%5C%7B%5Ccdot%5C%7D%5D" />. Here, the connections with free probability theory arise. In the relation to that eld we investigate the closed form of the asymptotic spectral distribution for the sum of the quadratic forms. Moreover, we put a free cumulant–moment relation formula that is based on the summation over partitions of the number. This formula is an alternative to the free cumulant{moment relation given through non{crossing partitions ofthe set. Furthermore, we investigate the normalized <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20E%5B%5Cprod_%7Bi=1%7D%5Ek%20Tr%5C%7BW%5E%7Bm_i%7D%5C%7D%5D" /> and derive, using the dierentiation with respect to some symmetric matrix, a recursive formula for that expectation. That allows us to re–establish moments of the Marcenko–Pastur distribution, and hence the recursive relation for the Catalan numbers. In this thesis we also prove that the <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20%5Cprod_%7Bi=1%7D%5Ek%20Tr%5C%7BW%5E%7Bm_i%7D%5C%7D" />, where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20W%5Csim%5Cmathcal%7BW%7D_p(I_p,n)" />, is a consistent estimator of the <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20E%5B%5Cprod_%7Bi=1%7D%5Ek%20Tr%5C%7BW%5E%7Bm_i%7D%5C%7D%5D" />. We consider <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20Y_t=%5Csqrt%7Bnp%7D%5Cbig(%5Cfrac%7B1%7D%7Bp%7DTr%5Cbig%5C%7B%5Cbig(%5Cfrac%7B1%7D%7Bn%7DW%5Cbig)%5Et%5Cbig%5C%7D-m%5E%7B(t)%7D_1%20(n,p)%5Cbig)," />, where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20m%5E%7B(t)%7D_1%20(n,p)=E%5Cbig%5B%5Cfrac%7B1%7D%7Bp%7DTr%5Cbig%5C%7B%5Cbig(%5Cfrac%7B1%7D%7Bn%7DW%5Cbig)%5Et%5Cbig%5C%7D%5Cbig%5D" />, which is proven to be normally distributed. Moreover, we propose, based on these random variables, a test for the identity of the covariance matrix using a goodness{of{t approach. The test performs very well regarding the power of the test compared to some presented alternatives for both the high–dimensional data (p &gt; n) and the multivariate data (p ≤ n).

Dans cette thèse, on aborde trois thèmes : problème de sélection de colonnes dans une matrice, distance de Banach-Mazur au cube et estimation de la covariance de matrices aléatoires. Bien que les trois thèmes paraissent éloignés, les techniques utilisées se ressemblent tout au long de la thèse. Dans un premier lieu, nous généralisons le principe d'invertibilité restreinte de Bourgain-Tzafriri. Ce résultat permet d'extraire un "grand" bloc de colonnes linéairement indépendantes dans une matrice et d'estimer la plus petite valeur singulière de la matrice extraite. Nous proposons ensuite un algorithme déterministe pour extraire d'une matrice un bloc presque isométrique c’est à dire une sous-matrice dont les valeurs singulières sont proches de 1. Ce résultat nous permet de retrouver le meilleur résultat connu sur la célèbre conjecture de Kadison-Singer. Des applications à la théorie locale des espaces de Banach ainsi qu'à l'analyse harmonique sont déduites. Nous donnons une estimation de la distance de Banach-Mazur d'un corps convexe de Rn au cube de dimension n. Nous proposons une démarche plus élémentaire, basée sur le principe d'invertibilité restreinte, pour améliorer et simplifier les résultats précédents concernant ce problème. Plusieurs travaux ont été consacrés pour approcher la matrice de covariance d'un vecteur aléatoire par la matrice de covariance empirique. Nous étendons ce problème à un cadre matriciel et on répond à la question. Notre résultat peut être interprété comme une quantification de la loi des grands nombres pour des matrices aléatoires symétriques semi-définies positives. L'estimation obtenue s'applique à une large classe de matrices aléatoires / In this thesis, we address three themes : columns subset selection in a matrix, the Banach-Mazur distance to the cube and the estimation of the covariance of random matrices. Although the three themes seem distant, the techniques used are similar throughout the thesis. In the first place, we generalize the restricted invertibility principle of Bougain-Tzafriri. This result allows us to extract a "large" block of linearly independent columns inside a matrix and estimate the smallest singular value of the restricted matrix. We also propose a deterministic algorithm in order to extract an almost isometric block inside a matrix i.e a submatrix whose singular values are close to 1. This result allows us to recover the best known result on the Kadison-Singer conjecture. Applications to the local theory of Banach spaces as well as to harmonic analysis are deduced. We give an estimate of the Banach-Mazur distance between a symmetric convex body in Rn and the cube of dimension n. We propose an elementary approach, based on the restricted invertibility principle, in order to improve and simplify the previous results dealing with this problem. Several studies have been devoted to approximate the covariance matrix of a random vector by its sample covariance matrix. We extend this problem to a matrix setting and we answer the question. Our result can be interpreted as a quantified law of large numbers for positive semidefinite random matrices. The estimate we obtain, applies to a large class of random matrices

by Foon-Yip Ng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 111-115). / Abstract also in Chinese. / Chapter Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Main Problems --- p.1 / Chapter 1.2 --- The Basic Concept of Decision Theory --- p.4 / Chapter 1.3 --- The Class of Orthogonally Invariant Estimators --- p.6 / Chapter 1.4 --- Related Works --- p.8 / Chapter 1.5 --- Summary --- p.10 / Chapter Chapter 2 --- Estimation of the Scale Matrix in a Wishart Distribution --- p.12 / Chapter 2.1 --- Review of the Previous Works --- p.13 / Chapter 2.2 --- Some Useful Statistical and Mathematical Results --- p.15 / Chapter 2.3 --- Improved Estimation of Σ under the Loss L1 --- p.18 / Chapter 2.4 --- Simulation Study for Wishart Distribution under the Loss L1 --- p.22 / Chapter 2.5 --- Improved Estimation of Σ under the Loss L2 --- p.25 / Chapter 2.6 --- Simulation Study for Wishart Distribution under the Loss L2 --- p.28 / Chapter Chapter 3 --- Estimation of the Scale Matrix in a Multivariate F Distribution --- p.31 / Chapter 3.1 --- Review of the Previous Works --- p.32 / Chapter 3.2 --- Some Useful Statistical and Mathematical Results --- p.35 / Chapter 3.3 --- Improved Estimation of Δ under the Loss L1____ --- p.38 / Chapter 3.4 --- Simulation Study for Multivariate F Distribution under the Loss L1 --- p.42 / Chapter 3.5 --- Improved Estimation of Δ under the Loss L2 ________ --- p.46 / Chapter 3.6 --- Relationship between Wishart Distribution and Multivariate F Distribution --- p.51 / Chapter 3.7 --- Simulation Study for Multivariate F Distribution under the Loss L2 --- p.52 / Chapter Chapter 4 --- Estimation of the Scale Matrix in an Elliptically Contoured Matrix Distribution --- p.57 / Chapter 4.1 --- Some Properties of Elliptically Contoured Matrix Distributions --- p.58 / Chapter 4.2 --- Review of the Previous Works --- p.60 / Chapter 4.3 --- Some Useful Statistical and Mathematical Results --- p.62 / Chapter 4.4 --- Improved Estimation of Σ under the Loss L3 --- p.63 / Chapter 4.5 --- Simulation Study for Multivariate-Elliptical t Distributions under the Loss L3 --- p.67 / Chapter 4.5.1 --- Properties of Multivariate-Elliptical t Distribution --- p.67 / Chapter 4.5.2 --- Simulation Study for Multivariate- Elliptical t Distributions --- p.70 / Chapter 4.6 --- Simulation Study for ε-Contaminated Normal Distributions under the Loss L3 --- p.74 / Chapter 4.6.1 --- Properties of ε-Contaminated Normal Distributions --- p.74 / Chapter 4.6.2 --- Simulation Study for 2-Contaminated Normal Distributions --- p.76 / Chapter 4.7 --- Discussions --- p.79 / APPENDIX --- p.81 / BIBLIOGRAPHY --- p.111

Made available in DSpace on 2014-06-11T19:22:18Z (GMT). No. of bitstreams: 0 Previous issue date: 2012Bitstream added on 2014-06-13T20:09:07Z : No. of bitstreams: 1 silva_glf_me_sjrp.pdf: 4891307 bytes, checksum: d50ac695507aa5097767c494c073e3f8 (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Neste trabalho estudaremos a relação existente entre polinômios ortogonais e matrizes aleatórias. Exibiremos uma caracterização de polinômios ortogonais via problemas de Riemann-Hilbert, a qual tem se mostrado uma ferramenta única para obtenção de assintóticas de polinômios ortogonais. Posteriormente, estudaremos a teoria básica dos ensembles unitários de matrizes aleatórias. Por fim, mostraremos como a teoria de assintóticas de polinômios ortogonais pode ser usada na análise assintótica de estatísticas de matrizes aleatórias, nos levando a resultados de universalidade para os ensembles unitários / We will exhibit a characterization of orthogonal p olynomials via Riemann-Hilbert problems, which has been shown a powerful to ol for studying asymptotics of orthogonal polynomials. Posteriorly we will review the basic theory of unitary ensembles of random matrices. At the end, we will show how asymptotics of orthogonal polynomials can be used to study asymptotics of several statistics in random matrix theory, obtaining universality results for the unitary ensembles

by Wai-Yin Chan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 42-45). / Chapter Chapter 1 --- Introduction / Chapter 1.1 --- Main Problems --- p.1 / Chapter 1.2 --- Class of Regularized Estimator --- p.4 / Chapter 1.3 --- Preliminaries --- p.6 / Chapter 1.4 --- Related Works --- p.9 / Chapter 1.5 --- Brief Summary --- p.10 / Chapter Chapter 2 --- Estimation of the Covariance Matrix and its Eigenvalues in a Wishart Distribution / Chapter 2.1 --- Significance of The Problem --- p.12 / Chapter 2.2 --- Review of the Previous Work --- p.13 / Chapter 2.3 --- Properties of the Wishart Distribution --- p.18 / Chapter 2.4 --- Main Results --- p.20 / Chapter 2.5 --- Simulation Study --- p.23 / Chapter Chapter 3 --- Estimation of the Scale Matrix and its Eigenvalues in a Multivariate F Distribution / Chapter 3.1 --- Formulation and significance of the Problem --- p.26 / Chapter 3.2 --- Review of the Previous Works --- p.28 / Chapter 3.3 --- Properties of Multivariate F Distribution --- p.30 / Chapter 3.4 --- Main Results --- p.33 / Chapter 3.5 --- Simulation Study --- p.38 / Chapter Chapter 4 --- Further research --- p.40 / Reference --- p.42 / Appendix --- p.46

In chapter 1, we present some background knowledge about random matrices, Coulomb gas, orthogonal polynomials, asymptotics of planar orthogonal polynomials and the Riemann-Hilbert problem. In chapter 2, we consider the monic orthogonal polynomials, $\{P_{n,N}(z)\}_{n=0,1,\cdots},$ that satisfy the orthogonality condition, \begin{equation}\nonumber \int_\mathbb{C}P_{n,N}(z)\overline{P_{m,N}(z)}e^{-N Q(z)}dA(z)=h_{n,N}\delta_{nm} \quad(n,m=0,1,2,\cdots), \end{equation} where $h_{n,N}$ is a (positive) norming constant and the external potential is given by $$Q(z)=|z|^2+ \frac{2c}{N}\log \frac{1}{|z-a|},\quad c>-1,\quad a>0.$$ The orthogonal polynomial is related to the interacting Coulomb particles with charge $+1$ for each, in the presence of an extra particle with charge $+c$ at $a.$ For $N$ large and a fixed ``c'' this can be a small perturbation of the Gaussian weight. The polynomial $P_{n,N}(z)$ can be characterized by a matrix Riemann--Hilbert problem \cite{Ba 2015}. We then apply the standard nonlinear steepest descent method \cite{Deift 1999, DKMVZ 1999} to derive the strong asymptotics of $P_{n,N}(z)$ when $n$ and $N$ go to $\infty.$ From the asymptotic behavior of $P_{n,N}(z),$ we find that, as we vary $c,$ the limiting distribution behaves discontinuously at $c=0.$ We observe that the mother body (a kind of potential theoretic skeleton) also behaves discontinuously at $c=0.$ The smooth interpolation of the discontinuity is obtained by further scaling of $c=e^{-\eta N}$ in terms of the parameter $\eta\in[0,\infty).$ To obtain the results for arbitrary values of $c$, we used the ``partial Schlesinger transform'' method developed in \cite{BL 2008} to derive an arbitrary order correction in the Riemann--Hilbert analysis. In chapter 3, we consider the case of multiple logarithmic singularities. The planar orthogonal polynomials $\{p_n(z)\}_{n=0,1,\cdots}$ with respect to the external potential that is given by $$Q(z)=|z|^2+ 2\sum_{j=1}^lc_j\log \frac{1}{|z-a_j|},$$ where $\{a_1, a_2, \cdots, a_l\}$ is a set of nonzero complex numbers and $\{c_1, c_2, \cdots, c_l\}$ is a set of positive real numbers. We show that the planar orthogonal polynomials $p_{n}(z)$ with $l$ logarithmic singularities in the potential are the multiple orthogonal polynomials $p_{{\bf{n}}}(z)$ (Hermite-Pad\'e polynomials) of Type II with $l$ measures of degree $|{\bf{n}}|=n=\kappa l+r,$ ${\bf{n}}=(n_1,\cdots,n_l)$ satisfying the orthogonality condition, $$ \frac{1}{2\ii}\int_{\Gamma}p_{{\bf{n}}}(z) z^k\chi_{{\bf{n}}-{\bf{e}}_j}(z)\dd z=0, \quad 0\leq k\leq n_j-1,\quad 1\leq j\leq l,$$ where $\Gamma$ is a certain simple closed curve with counterclockwise orientation and $$ \chi_{{\bf{n}}-{\bf{e}}_j}(z):= \prod_{i=1}^l(z-a_i)^{c_i }\int_{0}^{\overline{z}\times\infty}\frac{\prod_{i=1}^l(s-\bar{a}_i)^{n_i+c_i}}{(s-\bar{a}_j)\ee^{zs}}\,\dd s. $$ Such equivalence allows us to formulate the $(l+1)\times(l+1)$ Riemann--Hilbert problem for $p_n(z)$. We also find the ratio between the determinant of the moment matrix corresponding to the multiple orthogonal polynomials and the determinant of the moment matrix from the original planar measure.

In this thesis, we discuss some topics in random matrix theory which have applications to probability, statistics and quantum information theory. In Chapter 2, by relying on the spectral properties of an associated adjacency matrix, we find the distribution of the maximum of a Dyck path and show that it has the same distribution function as the unsigned Brownian excursion which was first derived in 1976 by Kennedy. We obtain a large and moderate deviation principle for the law of the maximum of a random Dyck path. Our result extends the results of Chung, Kennedy and Khorunzhiy and Marckert. In Chapter 3, we discuss a method of sampling called the Gibbs-slice sampler. This method is based on Neal's slice sampling combined with Gibbs sampling. In Chapter 4, we discuss several examples which have applications in physics and quantum information theory.

University of Macau / Faculty of Science and Technology / Department of Mathematics