The asymptotic distribution of the determinant of a random correlation matrix

TL;DR

This paper derives the asymptotic distribution of the determinant of a uniformly sampled random correlation matrix, providing insights relevant for statistical applications and theoretical understanding of matrix properties.

Contribution

It presents the first derivation of the asymptotic distribution of the determinant of a random correlation matrix sampled uniformly, linking it to broader random matrix theory.

Findings

Asymptotic distribution of the determinant derived

Connections made with the law of determinants in general random matrices

Several related results and implications discussed

Abstract

Random correlation matrices are studied for both theoretical interestingness and importance for applications. The author of [6] is interested in their interpretation as covariance matrices of purely random signals, the authors of [16] employ them in the generation of random clusters for studying clustering methods, whereas the authors of [8] use them for studying subset selection in multiple regression, etc. The determinant of a matrix is one of the most basic and important matrix functions, and this makes studying the distribution of the determinant of a random correlation matrix of paramount importance. Our main result gives the asymptotic distribution of the determinant of a random correlation matrix sampled from a uniform distribution over the space of $d \times d$ correlation matrices. Several spin-off results are proven along the way, and an interesting connection with the law of the determinant of general random matrices, proven in [15], is investigated.