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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2024
• Main Authors: Bagyan, Armine, Richards, Donald
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2024
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/212652
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions. Armine Bagyan and Donald Richards. SIGMA 20 (2024), 094, 22 pages

For positive integers and p such that ≥ , let ℝᵈˣᵖ denote the set of × real matrices, ₚ be the identity matrix of order , and d,ₚ = { ∈ ℝᵈˣᵖ ∣ ′ = ₚ} be the Stiefel manifold in ℝᵈˣᵖ. Complete asymptotic expansions as → ∞ are obtained for the normalizing constants of the matrix Bingham and matrix Langevin probability distributions on d,ₚ. The accuracy of each truncated expansion is strictly increasing in ; also, for sufficiently large , the accuracy is strictly increasing in , the number of terms in the truncated expansion. Lower bounds are obtained for the truncated expansions when the matrix parameters of the matrix Bingham distribution are positive definite and when the matrix parameter of the matrix Langevin distribution is of full rank. These results are applied to obtain the rates of convergence of the asymptotic expansions as both → ∞ and → ∞. Values of and arising in numerous data sets are used to illustrate the rate of convergence of the truncated approximations as or increases. These results extend recently obtained asymptotic expansions for the normalizing constants of the high-dimensional Bingham distributions.