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On the asymptotic spectral distribution of random matrices: closed form solutions using free independence
2013 (English)Licentiate thesis, monograph (Other academic)
Abstract [en]

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

$Q = \frac{1}n X_1X^\prime_1 + \cdot\cdot\cdot + \frac{1}n X_kX^\prime_k,$

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.

Series
Linköping studies in science and technology, ISSN 0280-7971 ; 1597
Keyword [en]
Spectral distribution, R-transform, Stieltjes transform, Free probability, Freeness, Asymptotic freeness
National Category
Probability Theory and Statistics
Identifiers
ISBN: 9789175195964 (print)OAI: oai:DiVA.org:lnu-58181DiVA: diva2:1047452
Presentation
2013-06-03, Planck, Fysikhuset, Campus Valla, 13:15 (English)
Supervisors
Available from: 2016-11-21 Created: 2016-11-17 Last updated: 2016-11-21Bibliographically approved

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Pielaszkiewicz, Jolanta Maria
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Probability Theory and Statistics