For positive integers $d$ and $p$ such that $d \ge p$, 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 $d, p \to \infty$. 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.

翻译：对于满足 $d \ge p$ 的正整数 $d$ 和 $p$，我们获得了Stiefel流形上矩阵Bingham分布和矩阵Langevin分布归一化常数在 $d$ 较大时的完全渐近展开式。每个截断展开的精度随 $d$ 严格递增；此外，对于充分大的 $d$，精度随截断展开项数 $m$ 严格递增。我们应用这些结果得到了当 $d, p \to \infty$ 时这些渐近展开的收敛速率。利用不同数据集中出现的 $d$ 和 $p$ 值，我们展示了截断近似随 $d$ 或 $m$ 增大时的收敛速率。这些结果扩展了我们近期关于高维Bingham分布归一化常数渐近展开的工作。