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Cumulant-moment relation in free probability theory
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Swedish University of Agricultural Sciences, Uppsala, Sweden.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.ORCID iD: 0000-0001-9896-4438
2014 (English)In: Acta et Commentationes Universitatis Tartuensis de Mathematica, ISSN 1406-2283, E-ISSN 2228-4699, Vol. 18, no 2, 265-278 p.Article in journal (Refereed) Published
##### Abstract [en]

The goal of this paper is to present and prove a cumulant-moment recurrent relation formula in free probability theory. It is convenient tool to determine underlying compactly supported distribution function. The existing recurrent relations between these objects require the combinatorial understanding of the idea of non-crossing partitions, which has been considered by Speicher and Nica. Furthermore, some formulations are given with additional use of the Möbius function. The recursive result derived in this paper does not require introducing any of those concepts. Similarly like the non-recursive formulation of Mottelson our formula demands only summing over partitions of the set. The proof of non-recurrent result is given with use of Lagrange inversion formula, while in our proof the calculations of the Stieltjes transform of the underlying measure are essential.

##### Place, publisher, year, edition, pages
University of Tartu Press , 2014. Vol. 18, no 2, 265-278 p.
##### Keyword [en]
R-transform, free cumulants, moments, free probability, non-commutative probability space, Stieltjes transform, random matrices
##### National Category
Probability Theory and Statistics Other Mathematics
##### Identifiers
OAI: oai:DiVA.org:liu-113087DiVA: diva2:777352
Available from: 2015-01-08 Created: 2015-01-08 Last updated: 2017-12-05Bibliographically 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
##### Identifiers
urn:nbn:se:liu:diva-122610 (URN)10.3384/diss.diva-122610 (DOI)978-91-7685-899-8 (ISBN)
##### Public defence
2015-12-11, Visionen, ingång 27, B-huset, Campus Valla, Linköping, 13:15 (English)
##### Supervisors
Available from: 2015-11-11 Created: 2015-11-11 Last updated: 2015-11-16Bibliographically approved

#### Open Access in DiVA

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#### Authority records BETA

Pielaszkiewicz, Jolantavon Rosen, DietrichSingull, Martin

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Cite
Citation style
• apa
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