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 ≥ 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 → ∞. 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.