Log determinant of large correlation matrices under infinite fourth moment

TL;DR

This paper proves a central limit theorem for the log determinant of large correlation matrices derived from heavy-tailed data with infinite fourth moments, extending understanding of high-dimensional random matrices.

Contribution

It establishes the CLT for the log determinant under infinite fourth moment conditions and symmetric heavy-tailed distributions, a novel extension in high-dimensional statistics.

Findings

CLT holds for heavy-tailed data with symmetric distributions and index <4.

Log determinant remains stable under infinite fourth moments.

Asymptotic normality fails if data lacks symmetry or has infinite third moments.

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.