For $d \ge 2$, let $X$ be a random vector having a Bingham distribution on $\mathcal{S}^{d-1}$, the unit sphere centered at the origin in $\R^d$, and let $\Sigma$ denote the symmetric matrix parameter of the distribution. Let $\Psi(\Sigma)$ be the normalizing constant of the distribution and let $\nabla \Psi_d(\Sigma)$ be the matrix of first-order partial derivatives of $\Psi(\Sigma)$ with respect to the entries of $\Sigma$. We derive complete asymptotic expansions for $\Psi(\Sigma)$ and $\nabla \Psi_d(\Sigma)$, as $d \to \infty$; these expansions are obtained subject to the growth condition that $\|\Sigma\|$, the Frobenius norm of $\Sigma$, satisfies $\|\Sigma\| \le \gamma_0 d^{r/2}$ for all $d$, where $\gamma_0 > 0$ and $r \in [0,1)$. Consequently, we obtain for the covariance matrix of $X$ an asymptotic expansion up to terms of arbitrary degree in $\Sigma$. Using a range of values of $d$ that have appeared in a variety of applications of high-dimensional spherical data analysis we tabulate the bounds on the remainder terms in the expansions of $\Psi(\Sigma)$ and $\nabla \Psi_d(\Sigma)$ and we demonstrate the rapid convergence of the bounds to zero as $r$ decreases.

翻译：对于 $d \ge 2$，设 $X$ 为服从 $\mathcal{S}^{d-1}$（$\R^d$ 中单位球面）上Bingham分布的随机向量，且令 $\Sigma$ 表示该分布的对称矩阵参数。设 $\Psi(\Sigma)$ 为分布的归一化常数，$\nabla \Psi_d(\Sigma)$ 为 $\Psi(\Sigma)$ 关于 $\Sigma$ 各元素的一阶偏导数矩阵。我们推导出当 $d \to \infty$ 时 $\Psi(\Sigma)$ 与 $\nabla \Psi_d(\Sigma)$ 的完全渐近展开；这些展开是在增长条件 $\|\Sigma\|$（$\Sigma$ 的Frobenius范数）对所有 $d$ 满足 $\|\Sigma\| \le \gamma_0 d^{r/2}$（其中 $\gamma_0 > 0$ 且 $r \in [0,1)$）下获得的。进而，我们得到 $X$ 的协方差矩阵关于 $\Sigma$ 任意阶项的渐近展开。通过使用高维球面数据分析中多种应用中出现的 $d$ 值范围，我们对 $\Psi(\Sigma)$ 与 $\nabla \Psi_d(\Sigma)$ 展开式中的余项进行数值界限列表，并验证了随着 $r$ 减小，这些界限迅速收敛至零。