Felix Seo: On phase transitions for the trace of squared sample correlation matrices in high dimension

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.