Topics in linear spectral statistics of random matrices

The behavior of the spectrum of a large random matrix is a topic of great interest in probability theory and statistics. At a global level, the limiting spectra of certain random matrix models have been known for some time. For example, the limiting spectral measure of a Wigner matrix is a semicircle law and the limiting spectral measure of a sample covariance matrix under certain conditions is a Marc̆enko-Pastur law. The local behavior of eigenvalues for specific random matrix ensembles (GUE and GOE) have been known for some time as well and until recently, were conjectured to be universal. There have been many recents breakthroughs in the universality of this local behavior of eigenvalues for Wigner Matrices. Furthermore, these universality results laws have been proven for other probabilistic models of particle systems, such as Beta Ensembles. In this thesis we investigate the fluctuations of linear statistics of eigenvalues of Wigner Matrices and Beta Ensembles in regimes intermediate to the global regime and the microscopic regime (called the mesoscopic regime). We verify that these fluctuations are Gaussian and derive the covariance for a range of test functions and scales. On a separate line of investigation, we study the global spectral behavior of a random matrix arising in statistics, called Kendall's Tau and verify that it satisfies an analogue of the Marc̆enko-Pastur Law.

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2017

Author dc:creator

• Lodhia, Asad

Advisor dc:contributor.advisor

• Alice Guionnet.

• Mathematics.

Statement dc:rights

• MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/112898

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/112898