In this thesis, we study the exact eigenvalue distribution of product of independent rectangular complex Gaussian matrices and also that of product of independent truncated Haar unitary matrices and inverses of truncated Haar unitary matrices. The eigenvalues of these random matrices form determinantal point processes on the complex plane. We also study the limiting expected empirical distribution of appropriately scaled eigenvalues of those matrices as the size of matrices go to infinity. We give the first example of a random matrix whose eigenvalues form a non-rotation invariant determinantal point process on the plane. The second theme of this thesis is infinite products of random matrices. We study the asymptotic behaviour of singular values and absolute values of eigenvalues of product of i .i .d matrices of fixed size, as the number of matrices in the product in-creases to infinity. In the special case of isotropic random matrices, We derive the asymptotic joint probability density of the singular values and also that of the absolute values of eigenvalues of product of right isotropic random matrices and show them to be equal. As a corollary of these results, we show probability that all the eigenvalues of product of certain i .i .d real random matrices of fixed size converges to one, as the number of matrices in the product increases to infinity.

by Lo Tai-yan, Milton. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 112-114). / Chapter 1 --- Preliminaries --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.1.1 --- The Noncentral Multivariate F distribution --- p.2 / Chapter 1.1.2 --- The Central Problems and the Approach --- p.4 / Chapter 1.2 --- Concepts and Terminology --- p.7 / Chapter 1.3 --- Choice of Estimates --- p.10 / Chapter 1.4 --- Related Work --- p.11 / Chapter 2 --- Estimation of the noncentrality parameter of a Noncentral Mul- tivariate F distribution --- p.19 / Chapter 2.1 --- Unbiased and Linear Estimators --- p.19 / Chapter 2.1.1 --- The unbiased estimate --- p.20 / Chapter 2.1.2 --- The Class of Linear Estimates --- p.24 / Chapter 2.2 --- Optimal Linear Estimate --- p.32 / Chapter 2.3 --- Nonlinear Estimate --- p.34 / Chapter 2.4 --- Monte Carlo Simulation Study --- p.41 / Chapter 2.5 --- Evaluation and Further Investigation --- p.42 / Chapter 3 --- Estimation of Canonical Correlation Coefficients --- p.73 / Chapter 3.1 --- Preliminary --- p.73 / Chapter 3.2 --- The Estimation Problem --- p.76 / Chapter 3.3 --- Orthogonally Invariant Estimates --- p.77 / Chapter 3.3.1 --- The Unbiased Estimate --- p.78 / Chapter 3.3.2 --- The Class of Linear Estimates --- p.78 / Chapter 3.3.3 --- The Class of Nonlinear Estimates --- p.80 / Chapter 3.4 --- Monte Carlo Simulation Study --- p.87 / Chapter 3.5 --- Evaluation and Further Investigation --- p.89 / Chapter A --- p.104 / Chapter A.1 --- Lemma 3.2 --- p.104 / Chapter A.2 --- Theorem 3.3 Leung(1992) --- p.105 / Chapter A.3 --- The Noncentral F Identity --- p.106 / Chapter B --- Bibliography --- p.111

Leung Kit Ying. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 86-88). / Abstracts in English and Chinese. / Declaration --- p.i / Acknowledgement --- p.ii / Chapter 1 --- INTRODUCTION --- p.1 / Chapter 2 --- IMPROVED ESTIMATION OF THE NORMAL PRECISION MATRIX USING THE L1 AND L2 LOSS FUNCTIONS --- p.7 / Chapter 2.1 --- Previous Work --- p.9 / Chapter 2.2 --- Important Lemmas --- p.13 / Chapter 2.3 --- Improved Estimation of Σ-1 under L1 Loss Function --- p.20 / Chapter 2.4 --- Improved Estimation of Σ-1 under L2 Loss Function --- p.26 / Chapter 2.5 --- Simulation Study --- p.31 / Chapter 2.6 --- Comparison with Krishnammorthy and Gupta's result --- p.38 / Chapter 3 --- IMPROVED ESTIMATION OF THE NORMAL PRECISION MATRIX USING THE L3 AND L4 LOSS FUNCTIONS --- p.43 / Chapter 3.1 --- Justification of the Loss Functions --- p.46 / Chapter 3.2 --- Important Lemmas for Calculating Risks --- p.48 / Chapter 3.3 --- Improved Estimation of Σ-1 under L3 Loss Function --- p.55 / Chapter 3.4 --- Improved Estimation of Σ-1 under L4 Loss Function --- p.62 / Chapter 3.5 --- Simulation Study --- p.69 / Appendix --- p.77 / Reference --- p.35

Il est bien connu que divers solides amorphes ont de nombreuses propriétés universelles. L'une d'entre elles est la variation de la conductivité thermique en fonction de la température. Cependant, le mécanisme microscopique du transfert de chaleur dans le domaine de température supérieure à 20 K est encore mal compris. Simulations numériques récentes du silicium et de la silice amorphes montrent que les modes de vibration dans la gamme de fréquences correspondante (au-dessus de plusieurs THz) sont délocalisés. En même temps ils sont complètement différents des phonons acoustiques de basse fréquence, dits « diffusions ».Dans ce travail, nous présentons un modèle stable de matrice aléatoire d'un solide amorphe. Dans ce modèle, on peut faire varier le degré de désordre allant du cristal parfait jusqu'au milieu mou extrêmement désordonné sans rigidité macroscopique. Nous montrons que les solides amorphes réels sont proches du deuxième cas limite, et que les diffusions occupent la partie dominante du spectre de vibration. La fréquence de transition entre les phonons acoustiques et diffusons est déterminée par le critère Ioffe-Regel. Fait intéressant, cette fréquence de transition coïncide pratiquement avec la position du pic Boson. Nous montrons également que la diffusivité et la densité d'états de vibration de diffusons sont pratiquement constantes en fonction de la fréquence. Par conséquent, la conductivité thermique est une fonction linéaire de la température dans le domaine allant à des températures relativement élevées, puis elle sature. Cette conclusion est en accord avec de nombreuses données expérimentales. En outre, nous considérons un modèle numérique de matériaux de type de silicium amorphe et étudions le rôle du désordre pour les vibrations longitudinales et transverses. Nous montrons aussi que la théorie des matrices aléatoires peut être appliquée avec succès pour estimer la densité d'états vibrationnels des systèmes granulaires bloqués. / It is well known that various amorphous solids have many universal properties. One of them is the temperature dependence of the thermal conductivity. However, the microscopic mechanism of the heat transfer above 20 K is still poorly understood. Recent numerical simulations of amorphous silicon and silica show that vibrational modes in the corresponding frequency range (above several THz) are delocalized, however they are completely different from low-frequency acoustic phonons, called “diffusons”.In this work we present a stable random matrix model of an amorphous solid. In this model one can vary the strength of disorder going from a perfect crystal to extremely disordered soft medium without macroscopic rigidity. We show that real amorphous solids are close to the second limiting case, and that diffusons occupy the dominant part of the vibrational spectrum. The crossover frequency between acoustic phonons and diffusons is determined by the Ioffe-Regel criterion. Interestingly, this crossover frequency practically coincides with the Boson peak position. We also show that, as a function of frequency, the diffusivity and the vibrational density of states of diffusons are practically constant. As a result, the thermal conductivity is a linear function of temperature up to rather high temperatures and then saturates. This conclusion is in agreement with numerous experimental data.Further, we consider a numerical model of amorphous silicon-like materials and investigate the role of disorder for longitudinal and transverse vibrations. We also show that the random matrix theory can be successfully applied to estimate the vibrational density of states of granular jammed systems.

We introduce real second-order freeness in second-order noncommutative probability spaces. We demonstrate that under this definition, independent ensembles of the three real models of random matrices which we consider, namely real Ginibre matrices, Gaussian orthogonal matrices, and real Wishart matrices, are asymptotically second-order free. These ensembles do not satisfy the complex definition of second-order freeness satisfied by their complex analogues. This definition may be used to calculate the asymptotic fluctuations of products of matrices in terms of the fluctuations of each ensemble. We use a combinatorial approach to the matrix calculations similar to genus expansion, but in which nonorientable surfaces appear, demonstrating the commonality between the real ensembles and the distinction from their complex analogues, motivating this distinct definition. We generalize the description of graphs on surfaces in terms of the symmetric group to the nonorientable case. In the real case we find, in addition to the terms appearing in the complex case corresponding to annular spoke diagrams, an extra set of terms corresponding to annular spoke diagrams in which the two circles of the annulus are oppositely oriented, and in which the matrix transpose appears. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-09-09 11:07:37.414

This thesis contains two main chapters. The first chapter focuses on the highdimensional asymptotic regimes of correlated Wishart matrices d−1YY^T , where Y is a n×d Gaussian random matrix with correlated and non-stationary entries. We provide quantitative bounds in the Wasserstein distance for the cases of central convergence and non-central convergence, verify such convergences hold in the weak topology of C([a; b]; M_n(R)), and show that our result can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. The second chapter develops a version of the Stein-Malliavin method in an infinite-dimensional and non-diffusive Poissonian setting. In particular, we provide quantitative central limit theorems for approximations by non-degenerate Hilbert-valued Gaussian random elements, as well as fourth moment bounds for approximating sequences with finite chaos expansion. We apply our results to the Brownian approximation of Poisson processes in Besov-Liouville spaces and also derive a functional limit theorem for an edge-counting statistic of a random geometric graph.

In the parametric development of statistical inference it often is assumed that observations are independent and Gaussian. The Gaussian assumption sometimes is justified on appeal to central limit theory or on the grounds that certain normal theory procedures are robust. The independence assumption, usually unjustified, routinely facilitates the derivation of needed distribution theory. In this thesis a variety of standard tests for scale parameters is considered when the observations are not necessarily either Gaussian or independent. The distributions considered are the spherically symmetric vector laws, i.e. laws for which x(nx1) and Px have the same distribution for every (nxn) orthogonal matrix P, and natural extensions of these to laws of random matrices. If x has a spherical law, then the distribution of Ax + b is said to be elliptically symmetric. The class of spherically symmetric laws contains such heavy-tailed distributions as the spherical Cauchy law and other symmetric stable distributions. As such laws need not have moments, the emphasis here is on tests for scale parameters which become tests regarding dispersion parameters whenever second-order moments are defined. Using the principle of invariance it is possible to characterize the invariant tests for certain hypotheses for all elliptically symmetric distributions. The particular problems treated are tests for the equality of k scale parameters, tests for the equality of k scale matrices, tests for sphericity, tests for block diagonal structure, tests for the uncorrelatedness of two variables within a set of m variables, and tests for the hypothesis of equi-correlatedness. In all cases except the last three the null and non-null distributions of invariant statistics are shown to be unique for all elliptically symmetric laws. The usual normal-theory procedures associated with these particular testing problems thus are exactly robust, and many of their known properties extend directly to this larger class. In the last three cases, the null distributions of certain invariant statistics are unique but the non-null distributions depend on the underlying elliptically symmetric law. In testing for block diagonal structure in the case of two blocks, a monotone power property is established for the subclass of all elliptically symmetric unimodal distributions. / Ph. D.

No description available.

Indiana University-Purdue University Indianapolis (IUPUI) / In this dissertation the partition function, $Z_n$, for the six-vertex model with domain wall boundary conditions is solved in the thermodynamic limit in various regions of the phase diagram. In the ferroelectric phase region, we show that $Z_n=CG^nF^{n^2}(1+O(e^{-n^{1-\ep}}))$ for any $\ep>0$, and we give explicit formulae for the numbers $C, G$, and $F$. On the critical line separating the ferroelectric and disordered phase regions, we show that $Z_n=Cn^{1/4}G^{\sqrt{n}}F^{n^2}(1+O(n^{-1/2}))$, and we give explicit formulae for the numbers $G$ and $F$. In this phase region, the value of the constant $C$ is unknown. In the antiferroelectric phase region, we show that $Z_n=C\th_4(n\om)F^{n^2}(1+O(n^{-1}))$, where $\th_4$ is Jacobi's theta function, and explicit formulae are given for the numbers $\om$ and $F$. The value of the constant $C$ is unknown in this phase region. In each case, the proof is based on reformulating $Z_n$ as the eigenvalue partition function for a random matrix ensemble (as observed by Paul Zinn-Justin), and evaluation of large $n$ asymptotics for a corresponding system of orthogonal polynomials. To deal with this problem in the antiferroelectric phase region, we consequently develop an asymptotic analysis, based on a Riemann-Hilbert approach, for orthogonal polynomials on an infinite regular lattice with respect to varying exponential weights. The general method and results of this analysis are given in Chapter 5 of this dissertation.

Compressed Sensing (CS) is an emerging paradigm in which signals are recovered from undersampled nonadaptive linear measurements taken at a rate proportional to the signal's true information content as opposed to its ambient dimension. The resulting problem consists in finding a sparse solution to an underdetermined system of linear equations. It has now been established, both theoretically and empirically, that certain optimization algorithms are able to solve such problems. Iterative Hard Thresholding (IHT) (Blumensath and Davies, 2007), which is the focus of this thesis, is an established CS recovery algorithm which is known to be effective in practice, both in terms of recovery performance and computational efficiency. However, theoretical analysis of IHT to date suffers from two drawbacks: state-of-the-art worst-case recovery conditions have not yet been quantified in terms of the sparsity/undersampling trade-off, and also there is a need for average-case analysis in order to understand the behaviour of the algorithm in practice. In this thesis, we present a new recovery analysis of IHT, which considers the fixed points of the algorithm. In the context of arbitrary matrices, we derive a condition guaranteeing convergence of IHT to a fixed point, and a condition guaranteeing that all fixed points are 'close' to the underlying signal. If both conditions are satisfied, signal recovery is therefore guaranteed. Next, we analyse these conditions in the case of Gaussian measurement matrices, exploiting the realistic average-case assumption that the underlying signal and measurement matrix are independent. We obtain asymptotic phase transitions in a proportional-dimensional framework, quantifying the sparsity/undersampling trade-off for which recovery is guaranteed. By generalizing the notion of xed points, we extend our analysis to the variable stepsize Normalised IHT (NIHT) (Blumensath and Davies, 2010). For both stepsize schemes, comparison with previous results within this framework shows a substantial quantitative improvement. We also extend our analysis to a related algorithm which exploits the assumption that the underlying signal exhibits tree-structured sparsity in a wavelet basis (Baraniuk et al., 2010). We obtain recovery conditions for Gaussian matrices in a simplified proportional-dimensional asymptotic, deriving bounds on the oversampling rate relative to the sparsity for which recovery is guaranteed. Our results, which are the first in the phase transition framework for tree-based CS, show a further significant improvement over results for the standard sparsity model. We also propose a dynamic programming algorithm which is guaranteed to compute an exact tree projection in low-order polynomial time.