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$p$，我们获得了在Stiefel流形上的矩阵Bingham分布和矩阵Langevin分布归一化常数在$d$趋于无穷时的完全渐近展开。每个截断展开的精度随$d$严格递增；此外，当$d$足够大时，精度随截断展开项数$m$严格递增。我们将这些结果应用于当$d, p \to \infty$时渐近展开收敛速度的确定。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$m$增大的收敛速度。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。