For positive integers $d$ and $p$ such that $d \ge p$, let $\mathbb{R}^{d \times p}$ denote the set of $d \times p$ real matrices, $I_p$ be the identity matrix of order $p$, and $V_{d,p} = \{x \in \mathbb{R}^{d \times p} \mid x'x = I_p\}$ be the Stiefel manifold in $\mathbb{R}^{d \times p}$. Complete asymptotic expansions as $d \to \infty$ are obtained for the normalizing constants of the matrix Bingham and matrix Langevin probability distributions on $V_{d,p}$. 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. 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 $d \to \infty$ and $p \to \infty$. Values of $d$ and $p$ arising in numerous data sets are used to illustrate the rate of convergence of the truncated approximations as $d$ or $m$ increases. These results extend recently-obtained asymptotic expansions for the normalizing constants of the high-dimensional Bingham distributions.

翻译：对于满足 $d \ge p$ 的正整数 $d$ 和 $p$，令 $\mathbb{R}^{d \times p}$ 表示 $d \times p$ 实矩阵的集合，$I_p$ 为 $p$ 阶单位矩阵，$V_{d,p} = \{x \in \mathbb{R}^{d \times p} \mid x'x = I_p\}$ 为 $\mathbb{R}^{d \times p}$ 中的 Stiefel 流形。本文获得了 $V_{d,p}$ 上的矩阵宾厄姆概率分布与矩阵朗之万概率分布的归一化常数在 $d \to \infty$ 时的完全渐近展开。每个截断展开的精度严格随 $d$ 增加而提高；此外，对于充分大的 $d$，其精度也严格随截断展开的项数 $m$ 增加而提高。当矩阵宾厄姆分布的矩阵参数为正定矩阵，以及当矩阵朗之万分布的矩阵参数为满秩矩阵时，本文获得了截断展开的下界。这些结果被应用于获取渐近展开在 $d \to \infty$ 与 $p \to \infty$ 同时发生时的收敛速率。通过众多数据集中出现的 $d$ 和 $p$ 值，说明了截断近似随 $d$ 或 $m$ 增加时的收敛速率。这些结果扩展了最近获得的高维宾厄姆分布归一化常数的渐近展开。