Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions

For positive integers $d$ and $p$ such that $d \ge p$, we obtain complete asymptotic expansions, for large $d$, of the normalizing constants for the matrix Bingham and matrix Langevin distributions on Stiefel manifolds. The accuracy of each truncated expansion is strictly increasing in $d$; also, for sufficiently large $d$, the accuracy is strictly increasing in $m$, the number of terms in the truncated expansion. We apply these results to obtain the rate of convergence of these asymptotic expansions if both $d, p \to \infty$. Using values of $d$ and $p$ arising in various data sets, we illustrate the rate of convergence of the truncated approximations as $d$ or $m$ increases. These results extend our recent work on asymptotic expansions for the normalizing constants of the high-dimensional Bingham distributions.