liu.seSearch for publications in DiVA
Change search

On the asymptotic spectral distribution of random matrices: Closed form solutions using free independence
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
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. Thesis, ISSN 0280-7971 ; 1597
##### Keyword [en]
spectral distribution, R-transform, Stieltjes transform, free probability, freeness, asymptotic freeness
##### National Category
Probability Theory and Statistics
##### Identifiers
Local ID: LIU-TEK-LIC-2013:31ISBN: 978-91-7519-596-4 (print)OAI: oai:DiVA.org:liu-92637DiVA: diva2:621444
##### Supervisors
Available from: 2013-05-22 Created: 2013-05-14 Last updated: 2014-09-29Bibliographically approved

#### Open Access in DiVA

On the asymptotic spectral distribution of random matrices: Closed form solutions using free independence(3432 kB)493 downloads
##### File information
File name FULLTEXT01.pdfFile size 3432 kBChecksum SHA-512
437b7443ce5c5342a3fd796b00046691367a0a6e0ddccf402919527fb7c26677fe716b46b06d2616fb261e5f117cdde9a5d2e392af171e4c7efcb329a19121d3
Type fulltextMimetype application/pdf
##### File information
File name COVER01.pdfFile size 211 kBChecksum SHA-512
e6ab2e8ddbbd85dddbaa04c726b8b9c5915dc2f41d91c396f563cc45b22feb18cd1b426269607e7edab4a72a106106dbf0e1e6565b853edd82786a0221c06783
Type coverMimetype application/pdf

#### Search in DiVA

##### By author/editor
Pielaszkiewicz, Jolanta
##### By organisation
Mathematical Statistics The Institute of Technology
##### On the subject
Probability Theory and Statistics