Complete Asymptotic Expansions and the High-Dimensional Bingham Distributions

Let $X$ denote a random vector having a Bingham distribution on ${\mathcal{S}}^{d-1}$, the unit sphere centered at the origin in ${\mathbb{R}}^d$. With $\Sigma$ denoting the symmetric matrix parameter of the distribution, let $\Psi(\Sigma)$ be the corresponding normalizing constant of the distribution. We derive for $\Psi(\Sigma)$ and its first-order partial derivatives complete asymptotic expansions as $d \to \infty$. These expansions are obtained under the growth condition that $\|\Sigma\|$, the Frobenius norm of $\Sigma$, satisfies $\|\Sigma\| = O(d^{r/2})$, where $0 \le r < 1$. As a consequence, we obtain for the covariance matrix of $X$ an asymptotic expansion up to terms of arbitrary degree in $\Sigma$.