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Closed Form of the Asymptotic Spectral Distribution of Random Matrices Using Free Independence
##### 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 $\mathcal{Q}$, which can be decomposed into the sum of asymptotically free independent summands.

Let $(\Omega,\mathcal{F},P)$ be a probability space. We consider the particular example of a non-commutative space$(RM_p(\mathbb{C}),\tau)$, where $RM_p(\mathbb{C})$ denotes the set of all  $p \times p$ random matrices, with entries which are com-plex random variables with finite moments of any order and $\tau$ 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$Q\in\mathcal{Q}$.

Finally, the main result is a new theorem pointing out classes of the matrix $Q$ 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.

##### Series
LiTH-MAT-R, ISSN 0348-2960 ; 2015:12
##### Keyword [en]
Closed form solutions, Free probability, Spectral distribution, Asymptotic, Random matrices, Free independence
Mathematics
##### Identifiers
OAI: oai:DiVA.org:lnu-58165DiVA, id: diva2:1047444
Available from: 2016-11-17 Created: 2016-11-17 Last updated: 2018-04-20Bibliographically 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 $\small\frac{p}{n}$ 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 $\small \frac{1}{p}E[Tr\{\cdot\}]$. 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 $\small E[\prod_{i=1}^k Tr\{W^{m_i}\}]$ 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 $\small \prod_{i=1}^k Tr\{W^{m_i}\}$, where $\small W\sim\mathcal{W}_p(I_p,n)$, is a consistent estimator of the $\small E[\prod_{i=1}^k Tr\{W^{m_i}\}]$. We consider

$\small Y_t=\sqrt{np}\big(\frac{1}{p}Tr\big\{\big(\frac{1}{n}W\big)^t\big\}-m^{(t)}_1 (n,p)\big),$,

where $\small m^{(t)}_1 (n,p)=E\big[\frac{1}{p}Tr\big\{\big(\frac{1}{n}W\big)^t\big\}\big]$, 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).

##### Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1724
##### Keyword
Eigenvalue distribution, free moments, free Poisson law, Marchenko-Pastur law, random matrices, spectral distribution, Wishart matrix
Mathematics
##### Research subject
Natural Science, Mathematics
##### Identifiers
urn:nbn:se:lnu:diva-58164 (URN)10.3384/diss.diva-122610 (DOI)978-91-7685-899-8 (ISBN)
##### Public defence
2015-12-11, Visionen, ingång 27, B-huset, 13:15 (English)
##### Supervisors
Available from: 2016-11-18 Created: 2016-11-17 Last updated: 2016-11-21Bibliographically approved

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Pielaszkiewicz, Jolanta MariaSingull, Martin
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Cite
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