Log determinant of large correlation matrices under infinite fourth moment

Abstract

In this paper, we show the central limit theorem for the logarithmic determinant of the sample correlation matrix $\mathbf{R}$ constructed from the $(p\times n)$-dimensional data matrix $\mathbf{X}$ containing independent and identically distributed random entries with mean zero, variance one and infinite fourth moments. Precisely, we show that for $p/n\to \gamma \in (0,1)$ as $n,p\to \infty$ the logarithmic law \begin{equation*} \frac{\log \det \mathbf{R} -(p-n+\frac{1}{2})\log(1-p/n)+p-p/n}{\sqrt{-2\log(1-p/n)- 2 p/n}} \overset{d}{\rightarrow} N(0,1)\, \end{equation*} is still valid if the entries of the data matrix $\mathbf{X}$ follow a symmetric distribution with a regularly varying tail of index $\alpha \in (3,4)$. The latter assumptions seem to be crucial, which is justified by the simulations: if the entries of $\mathbf{X}$ have the infinite absolute third moment and/or their distribution is not symmetric, the logarithmic law is not valid anymore. The derived results highlight that the logarithmic determinant of the sample correlation matrix is a very stable and flexible statistic for heavy-tailed big data and open a novel way of analysis of high-dimensional random matrices with self-normalized entries.

28 pages, 2 figures. This is an old verison. A revised version appears in Annales de l'Institut Henri Poincaré - Probabilités et Statistiques (2023)