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 → ∞) 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 = Γ ΔΘ′ of an m × k parameter matrix F, where Γ ∈ Vp,m, Θ ∈ Vp,k, and Δ = diag(λ1, ..., λp), λj > 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 = m1/2Γ′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 Δ = 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.