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).

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).

It was conjectured in the early 1950¡¯s that the empirical spectral distribution (ESD) of an $n \times n$ matrix whose entries are independent and identically distributed with mean zero and variance one, normalized by a factor of $\frac{1}{\sqrt{n}}$, converges to the uniform distribution over the unit disk on the complex plane, which is called the circular law. The goal of this thesis is to prove the so called Replacement Principle introduced by Tao and Vu which is a crucial step in their recent proof of the circular law in full generality. It gives a general criterion for the difference of the ESDs of two normalised random matrices $\frac{1}{\sqrt{n}}A_n$, $\frac{1}{\sqrt{n}}B_n$ to converge to 0. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2009-07-11 14:57:44.225

The spectral distribution function of random matrices is an information-carrying object widely studied within Random matrix theory. In this thesis we combine the results of the theory together with the idea of free independence introduced by Voiculescu (1985). Important theoretical part of the thesis consists of the introduction to Free probability theory, which justifies use of asymptotic freeness with respect to particular matrices as well as the use of Stieltjes and R-transform. Both transforms are presented together with their properties. The aim of thesis is to point out characterizations of those classes of the matrices, which have closed form expressions for the asymptotic spectral distribution function. We consider all matrices which can be decomposed to the sum of asymptotically free independent summands. In particular, explicit calculations are performed in order to illustrate the use of asymptotic free independence to obtain the asymptotic spectral distribution for a matrix Q and generalize Marcenko and Pastur (1967) theorem. The matrix Q is defined as <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?Q%20=%20%5Cfrac%7B1%7Dn%20X_1X%5E%5Cprime_1%20+%20%5Ccdot%5Ccdot%5Ccdot%20+%20%5Cfrac%7B1%7Dn%20X_kX%5E%5Cprime_k," />  where Xi is p × n matrix following a matrix normal distribution, Xi ~ Np,n(0, \sigma^2I, I). Finally, theorems pointing out classes of matrices Q which lead to closed formula for the asymptotic spectral distribution will be presented. Particularly, results for matrices with inverse Stieltjes transform, with respect to the composition, given by a ratio of polynomials of 1st and 2nd degree, are given.

The spectral distribution function of random matrices is an information-carrying object widely studied within Random matrix theory. In this thesis we combine the results of the theory together with the idea of free independence introduced by Voiculescu (1985). Important theoretical part of the thesis consists of the introduction to Free probability theory, which justifies use of asymptotic freeness with respect to particular matrices as well as the use of Stieltjes and R-transform. Both transforms are presented together with their properties. The aim of thesis is to point out characterizations of those classes of the matrices, which have closed form expressions for the asymptotic spectral distribution function. We consider all matrices which can be decomposed to the sum of asymptotically free independent summands. In particular, explicit calculations are performed in order to illustrate the use of asymptotic free independence to obtain the asymptotic spectral distribution for a matrix Q and generalize Marcenko and Pastur (1967) theorem. The matrix Q is defined as <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?Q%20=%20%5Cfrac%7B1%7Dn%20X_1X%5E%5Cprime_1%20+%20%5Ccdot%5Ccdot%5Ccdot%20+%20%5Cfrac%7B1%7Dn%20X_kX%5E%5Cprime_k," />  where Xi is p × n matrix following a matrix normal distribution, Xi ~ Np,n(0, \sigma^2I, I). Finally, theorems pointing out classes of matrices Q which lead to closed formula for the asymptotic spectral distribution will be presented. Particularly, results for matrices with inverse Stieltjes transform, with respect to the composition, given by a ratio of polynomials of 1st and 2nd degree, are given.

This dissertation has two parts. In the first part, we focus on random inner product kernel matrices. Under various assumptions, many authors have proved that the limiting empirical spectral distribution (ESD) of such matrices A converges to the Marchenko- Pastur distribution. Here, we establish the corresponding rate of convergence. The strategy is as follows. First, we show that for z = u + iv ∈ C, v &gt; 0, the distance between the Stieltjes transform m_A (z) of ESD of matrix A and Machenko-Pastur distribution m(z) is of order O (log n \ nv). Next, we prove the Kolmogorov distance between ESD of matrix A and Marchenko-Pastur distribution is of order O(3\log n\n). It is the less sharp rate for much more general class of matrices. This uses a Berry-Esseen type bound that has been employed for similar purposes for other families of random matrices. In the second part, random geometric graphs on the unit sphere are considered. Observing that adjacency matrices of these graphs can be thought of as random inner product matrices, we are able to use an idea of Cheng-Singer to establish the limiting for the ESD of these adjacency matrices. / Mathematics

Thesis briefly discusses the issues of color perception and effects associated with it. Further describes color model and its mathematical definition, which are used by color management. Briefly analyzes important elements of ICC profile. In second part two java applications were designed and programmed. First one evaluates visible spectrum and graphically demonstrate procedure for obtaining trichromacy information from this spectrum. Second application analyzes ICC profile and derives gamut of described device.

Cette thèse porte essentiellement sur l'étude de la distribution spectrale limite de grandes matrices aléatoires dont les entrées sont corrélées et traite également d'inégalités de déviation pour la plus grande valeur propre d'une somme de matrices aléatoires auto-adjointes et géométriquement absolument réguliers. On s'intéresse au comportement asymptotique de grandes matrices de covariances et de matrices de type Wigner dont les entrées sont des fonctionnelles d'une suite de variables aléatoires à valeurs réelles indépendantes et de même loi. On montre que dans ce contexte la distribution spectrale empirique des matrices peut être obtenue en analysant une matrice gaussienne ayant la même structure de covariance. Cette approche est valide que ce soit pour des processus à mémoire courte ou pour des processus exhibant de la mémoire longue, et on montre ainsi un résultat d'universalité concernant le comportement asymptotique du spectre de ces matrices. Notre approche consiste en un mélange de la méthode de Lindeberg par blocs et d'une technique d'interpolation Gaussienne. Une nouvelle inégalité de concentration pour la transformée de Stieltjes pour des matrices symétriques ayant des lignes $m$-dépendantes est établie. Notre méthode permet d'obtenir, sous de faibles conditions, l'équation intégrale satisfaite par la transformée de Stieltjes de la distribution spectrale limite. Ce résultat s'applique à des matrices associées à des fonctions de processus linéaires, à des modèles ARCH ainsi qu'à des modèles non-linéaires de type Volterra. On traite également le cas des matrices de Gram dont les entrées sont des fonctionnelles d'un processus absolument régulier (i.e. $beta$-mélangeant).On établit une inégalité de concentration qui nous permet de montrer, sous une condition de décroissance arithmétique des coefficients de $beta$-mélange, que la transformée de Stieltjes se concentre autour de sa moyenne. On réduit ensuite le problème à l'étude d'une matrice gaussienne ayant une structure de covariance similaire via la méthode de Lindeberg par blocs. Des applications à des chaînes de Markov stationnaires et Harris récurrentes ainsi qu'à des systèmes dynamiques sont données. Dans le dernier chapitre de cette thèse, on étudie des inégalités de déviation pour la plus grande valeur propre d'une somme de matrices aléatoires auto-adjointes. Plus précisément, on établit une inégalité de type Bernstein pour la plus grande valeur propre de la somme de matrices auto-ajointes, centrées et géométriquement $beta$-mélangeantes dont la plus grande valeur propre est bornée. Ceci étend d'une part le résultat de Merlevède et al. (2009) à un cadre matriciel et généralise d'autre part, à un facteur logarithmique près, les résultats de Tropp (2012) pour des sommes de matrices indépendantes / In this thesis, we investigate mainly the limiting spectral distribution of random matrices having correlated entries and prove as well a Bernstein-type inequality for the largest eigenvalue of the sum of self-adjoint random matrices that are geometrically absolutely regular. We are interested in the asymptotic spectral behavior of sample covariance matrices and Wigner-type matrices having correlated entries that are functions of independent random variables. We show that the limiting spectral distribution can be obtained by analyzing a Gaussian matrix having the same covariance structure. This approximation approach is valid for both short and long range dependent stationary random processes just having moments of second order. Our approach is based on a blend of a blocking procedure, Lindeberg's method and the Gaussian interpolation technique. We also develop new tools including a concentration inequality for the spectral measure for matrices having $K$-dependent rows. This method permits to derive, under mild conditions, an integral equation of the Stieltjes transform of the limiting spectral distribution. Applications to matrices whose entries consist of functions of linear processes, ARCH processes or non-linear Volterra-type processes are also given.We also investigate the asymptotic behavior of Gram matrices having correlated entries that are functions of an absolutely regular random process. We give a concentration inequality of the Stieltjes transform and prove that, under an arithmetical decay condition on the absolute regular coefficients, it is almost surely concentrated around its expectation. The study is then reduced to Gaussian matrices, with a close covariance structure, proving then the universality of the limiting spectral distribution. Applications to stationary Harris recurrent Markov chains and to dynamical systems are also given.In the last chapter, we prove a Bernstein type inequality for the largest eigenvalue of the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality is an extension to the matrix setting of the Bernstein-type inequality obtained by Merlev`ede et al. (2009) and a generalization, up to a logarithmic term, of Tropp's inequality (2012) by relaxing the independence hypothesis

Cette thèse porte essentiellement sur l'étude de la distribution spectrale limite de grandes matrices aléatoires dont les entrées sont corrélées et traite également d'inégalités de déviation pour la plus grande valeur propre d'une somme de matrices aléatoires auto-adjointes et géométriquement absolument réguliers. On s'intéresse au comportement asymptotique de grandes matrices de covariances et de matrices de type Wigner dont les entrées sont des fonctionnelles d'une suite de variables aléatoires à valeurs réelles indépendantes et de même loi. On montre que dans ce contexte la distribution spectrale empirique des matrices peut être obtenue en analysant une matrice gaussienne ayant la même structure de covariance. Cette approche est valide que ce soit pour des processus à mémoire courte ou pour des processus exhibant de la mémoire longue, et on montre ainsi un résultat d'universalité concernant le comportement asymptotique du spectre de ces matrices. Notre approche consiste en un mélange de la méthode de Lindeberg par blocs et d'une technique d'interpolation Gaussienne. Une nouvelle inégalité de concentration pour la transformée de Stieltjes pour des matrices symétriques ayant des lignes $m$-dépendantes est établie. Notre méthode permet d'obtenir, sous de faibles conditions, l'équation intégrale satisfaite par la transformée de Stieltjes de la distribution spectrale limite. Ce résultat s'applique à des matrices associées à des fonctions de processus linéaires, à des modèles ARCH ainsi qu'à des modèles non-linéaires de type Volterra. On traite également le cas des matrices de Gram dont les entrées sont des fonctionnelles d'un processus absolument régulier (i.e. $beta$-mélangeant).On établit une inégalité de concentration qui nous permet de montrer, sous une condition de décroissance arithmétique des coefficients de $beta$-mélange, que la transformée de Stieltjes se concentre autour de sa moyenne. On réduit ensuite le problème à l'étude d'une matrice gaussienne ayant une structure de covariance similaire via la méthode de Lindeberg par blocs. Des applications à des chaînes de Markov stationnaires et Harris récurrentes ainsi qu'à des systèmes dynamiques sont données. Dans le dernier chapitre de cette thèse, on étudie des inégalités de déviation pour la plus grande valeur propre d'une somme de matrices aléatoires auto-adjointes. Plus précisément, on établit une inégalité de type Bernstein pour la plus grande valeur propre de la somme de matrices auto-ajointes, centrées et géométriquement $beta$-mélangeantes dont la plus grande valeur propre est bornée. Ceci étend d'une part le résultat de Merlevède et al. (2009) à un cadre matriciel et généralise d'autre part, à un facteur logarithmique près, les résultats de Tropp (2012) pour des sommes de matrices indépendantes / In this thesis, we investigate mainly the limiting spectral distribution of random matrices having correlated entries and prove as well a Bernstein-type inequality for the largest eigenvalue of the sum of self-adjoint random matrices that are geometrically absolutely regular. We are interested in the asymptotic spectral behavior of sample covariance matrices and Wigner-type matrices having correlated entries that are functions of independent random variables. We show that the limiting spectral distribution can be obtained by analyzing a Gaussian matrix having the same covariance structure. This approximation approach is valid for both short and long range dependent stationary random processes just having moments of second order. Our approach is based on a blend of a blocking procedure, Lindeberg's method and the Gaussian interpolation technique. We also develop new tools including a concentration inequality for the spectral measure for matrices having $K$-dependent rows. This method permits to derive, under mild conditions, an integral equation of the Stieltjes transform of the limiting spectral distribution. Applications to matrices whose entries consist of functions of linear processes, ARCH processes or non-linear Volterra-type processes are also given.We also investigate the asymptotic behavior of Gram matrices having correlated entries that are functions of an absolutely regular random process. We give a concentration inequality of the Stieltjes transform and prove that, under an arithmetical decay condition on the absolute regular coefficients, it is almost surely concentrated around its expectation. The study is then reduced to Gaussian matrices, with a close covariance structure, proving then the universality of the limiting spectral distribution. Applications to stationary Harris recurrent Markov chains and to dynamical systems are also given.In the last chapter, we prove a Bernstein type inequality for the largest eigenvalue of the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality is an extension to the matrix setting of the Bernstein-type inequality obtained by Merlev`ede et al. (2009) and a generalization, up to a logarithmic term, of Tropp's inequality (2012) by relaxing the independence hypothesis