Complete asymptotic expansions and the high-dimensional Bingham distributions

For $$d \ge 2$$ d ≥ 2 , let X be a random vector having a Bingham distribution on $${\mathcal {S}}^{d-1}$$ S d - 1 , the unit sphere centered at the origin in $${\mathbb {R}}^d$$ R d , and let $$\Sigma $$ Σ denote the symmetric matrix parameter of the distribution. Let $$\Psi (\Sigma )$$ Ψ ( Σ ) be the normalizing constant of the distribution and let $$\nabla \Psi _d(\Sigma )$$ ∇ Ψ d ( Σ ) be the matrix of first-order partial derivatives of $$\Psi (\Sigma )$$ Ψ ( Σ ) with respect to the entries of $$\Sigma $$ Σ . We derive complete asymptotic expansions for $$\Psi (\Sigma )$$ Ψ ( Σ ) and $$\nabla \Psi _d(\Sigma )$$ ∇ Ψ d ( Σ ) , as $$d \rightarrow \infty $$ d → ∞ ; these expansions are obtained subject to the growth condition that $$\Vert \Sigma \Vert $$ ‖ Σ ‖ , the Frobenius norm of $$\Sigma $$ Σ , satisfies $$\Vert \Sigma \Vert \le \gamma _0 d^{r/2}$$ ‖ Σ ‖ ≤ γ 0 d r / 2 for all d, where $$\gamma _0 > 0$$ γ 0 > 0 and $$r \in [0,1)$$ r ∈ [ 0 , 1 ) . Consequently, we obtain for the covariance matrix of X an asymptotic expansion up to terms of arbitrary degree in $$\Sigma $$ Σ . 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 $$\Psi (\Sigma )$$ Ψ ( Σ ) and $$\nabla \Psi _d(\Sigma )$$ ∇ Ψ d ( Σ ) and we demonstrate the rapid convergence of the bounds to zero as r decreases.