High-dimensional asymptotic behavior of the difference between the log-determinants of two Wishart matrices

In this paper, we evaluate the asymptotic behavior of the difference between the log-determinants of two random matrices distributed according to the Wishart distribution by using a high-dimensional asymptotic framework in which the size of the matrices and the degrees of freedom both approach infinity simultaneously. We consider two cases, depending whether a matrix is completely or partially included in another matrix. From the asymptotic behavior, we derive the condition needed to ensure consistency for a given log-likelihood-based information criterion for selecting variables in a canonical correlation analysis.