The asymptotic distribution of the determinant of a random correlation matrix

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.