liu.seSearch for publications in DiVA
Change search
ReferencesLink to record
Permanent link

Direct link
Closed Form of the Asymptotic Spectral Distribution of Random Matrices Using Free Independence
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0001-9896-4438
2015 (English)Report (Other academic)
Abstract [en]

The spectral distribution function of random matrices is an information-carrying object widely studied within Random matrix theory. Random matrix theory is the main eld placing its research interest in the diverse properties of matrices, with a particular emphasis placed on eigenvalue distribution. The aim of this article is to point out some classes of matrices, which have closed form expressions for the asymptotic spectral distribution function. We consider matrices, later denoted by , which can be decomposed into the sum of asymptotically free independent summands.

Let  be a probability space. We consider the particular example of a non-commutative space, where  denotes the set of all   random matrices, with entries which are com-plex random variables with finite moments of any order and  is tracial functional. In particular, explicit calculations are performed in order to generalize the theorem given in [15] and illustrate the use of asymptotic free independence to obtain the asymptotic spectral distribution for a particular form of matrix.

Finally, the main result is a new theorem pointing out classes of the matrix  which leads to a closed formula for the asymptotic spectral distribution. Formulation of results for matrices with inverse Stieltjes transforms, with respect to the composition, given by a ratio of 1st and 2nd degree polynomials, is provided.

Place, publisher, year, edition, pages
Linköping University Electronic Press, 2015. , 25 p.
LiTH-MAT-R, ISSN 0348-2960 ; 2015:12
Keyword [en]
closed form solutions, free probability, spectral distribution, asymptotic, random matrices, free independence.
National Category
URN: urn:nbn:se:liu:diva-122170ISRN: LiTH-MAT-R--2015/12--SEOAI: diva2:862644
Available from: 2015-10-23 Created: 2015-10-23 Last updated: 2015-11-12Bibliographically approved
In thesis
1. Contributions to High–Dimensional Analysis under Kolmogorov Condition
Open this publication in new window or tab >>Contributions to High–Dimensional Analysis under Kolmogorov Condition
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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 > n, assuming that the ratio  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 . 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  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 , where , is a consistent estimator of the . We consider


where , 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 > n) and the multivariate data (p ≤ n).

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. 61 p.
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1724
Eigenvalue distribution; free moments; free Poisson law; Marchenko-Pastur law; random matrices; spectral distribution; Wishart matrix.
National Category
urn:nbn:se:liu:diva-122610 (URN)10.3384/diss.diva-122610 (DOI)978-91-7685-899-8 (print) (ISBN)
Public defence
2015-12-11, Visionen, ingång 27, B-huset, Campus Valla, Linköping, 13:15 (English)
Available from: 2015-11-11 Created: 2015-11-11 Last updated: 2015-11-16Bibliographically approved

Open Access in DiVA

Closed Form of the Asymptotic Spectral Distribution of Random Matrices Using Free Independence(706 kB)85 downloads
File information
File name FULLTEXT01.pdfFile size 706 kBChecksum SHA-512
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Pielaszkiewicz, JolantaSingull, Martin
By organisation
Mathematical Statistics Faculty of Science & Engineering

Search outside of DiVA

GoogleGoogle Scholar
Total: 85 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Total: 132 hits
ReferencesLink to record
Permanent link

Direct link