Contributions to High–Dimensional Analysis under Kolmogorov Condition

University dissertation from Linköping : Linköping University Electronic Press

Author: Jolanta Maria Pielaszkiewicz; Linköpings Universitet.; Linköpings Universitet.; [2015]

Keywords: Natural Sciences Mathematics; Naturvetenskap Matematik; Eigenvalue distribution; free moments; free Poisson law; Marchenko-Pastur law; random matrices; spectral distribution; Wishart matrix.;

Abstract: 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 $\frac{p}{n}$ 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 $\frac{1}{p}E[Tr\{\cdot\}]$ . 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 $E[\prod_{i=1}^k Tr\{W^{m_i}\}]$ 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 $\prod_{i=1}^k Tr\{W^{m_i}\}$ , where $W\sim\mathcal{W}_p(I_p,n)$ , is a consistent estimator of the $E[\prod_{i=1}^k Tr\{W^{m_i}\}]$ . We consider $Y_t=\sqrt{np}\big(\frac{1}{p}Tr\big\{\big(\frac{1}{n}W\big)^t\big\}-m^{(t)}_1 (n,p)\big),$ ,where $m^{(t)}_1 (n,p)=E\big[\frac{1}{p}Tr\big\{\big(\frac{1}{n}W\big)^t\big\}\big]$ , 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).