High Dimensional Asymptotic Expansions for the Matrix Langevin Distributions on the Stiefel Manifold

Let Vk,m denote the Stiefel manifold whose elements are m - k (m >= k) matrices X such that X'X = Ik. We may be interested in high dimensional (as m --> [infinity]) asymptotic behaviors of statistics on Vk,m. High dimensional Stiefel manifolds may appear in a geometrical study in other contexts, e.g., for the analysis of compositional data with an arbitrary number m of components. We consider the matrix Langevin L(m, k; F) and L(m, k; m1/2F) distributions, each with the singular value decomposition F = [Gamma] [Delta][Theta]' of an m - k parameter matrix F, where [Gamma] [set membership, variant] Vp,m, [Theta] [set membership, variant] Vp,k, and [Delta] = diag([lambda]1, ..., [lambda]p), [lambda]j > 0. For a random matrix X having each of the two distributions, we derive asymptotic expansions, for large m, for the probability density functions of the matrix variates Y = m1/2[Gamma]'X and W = YY' and of the related functions y = tr MY' /(tr MM')1/2 and w = tr W. Here M is an arbitrary p - k constant matrix. Putting [Delta] = 0 in the asymptotic expansions yields those for the uniform distribution. The asymptotic expansions derived in this paper may be useful for statistical inference on Vk,m.