An Introduction to Wishart Matrix Moments
Abstract
These lecture notes provide a comprehensive, selfcontained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory, or as a selfcontained exposition of Wishart matrix moments. Random matrix theory plays a central role in statistical physics, computational mathematics and engineering sciences, including data assimilation, signal processing, combinatorial optimization, compressed sensing, econometrics and mathematical finance, among numerous others. The mathematical foundations of the theory of random matrices lies at the intersection of combinatorics, noncommutative algebra, geometry, multivariate functional and spectral analysis, and of course statistics and probability theory. As a result, most of the classical topics in random matrix theory are technical, and mathematically difficult to penetrate for nonexperts and regular users and practitioners. The technical aim of these notes is to review and extend some important results in random matrix theory in the specific context of real random Wishart matrices. This special class of Gaussiantype sample covariance matrix plays an important role in multivariate analysis and in statistical theory. We derive nonasymptotic formulae for the full matrix moments of real valued Wishart random matrices. As a corollary, we derive and extend a number of spectral and tracetype results for the case of nonisotropic Wishart random matrices. We also derive the full matrix moment analogues of some classic spectral and tracetype moment results. For example, we derive semicircle and MarchenckoPasturtype laws in the nonisotropic and full matrix cases. Laplace matrix transforms and matrix moment estimates are also studied, along with new spectral and trace concentrationtype inequalities.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 arXiv:
 arXiv:1710.10864
 Bibcode:
 2017arXiv171010864B
 Keywords:

 Mathematics  Probability;
 Mathematics  Combinatorics
 EPrint:
 Foundations and Trends in Machine Learning, Volume 11, No. 2, pages: 97218, (2018)