Felix Seo: On phase transitions for the trace of squared sample correlation matrices in high dimension
Master Thesis
Time: Wed 2024-06-05 11.25 - 12.05
Location: Meeting room 9, floor 2, house 1, Albano
Respondent: Felix Seo
Supervisor: Johannes Heiny
Abstract.
We provide limit theory for the trace of the squared sample correlation matrix \(R\), constructed from \(n\) observations of a \(p\)-dimensional random vector with iid components. If the entries have finite fourth moment and \(p\) and \(n\) grow proportionally, it is known that \(\operatorname{tr}(R^2)\) satisfies a central limit theorem (CLT) and the centering and scaling sequences are universal in the sense that they do not depend on the entry distribution. Under a symmetry and a regular variation assumption with index \(\alpha\) and any growth rate of the dimension, we prove that the universal CLT remains valid for \(\alpha > 3\). Moreover, for \(\alpha \le 3\) we establish a non-universal CLT with norming sequences depending on the value of \(\alpha\). Our findings are illustrated in a small simulation study.