Complete Asymptotic Expansions and the High-Dimensional Bingham Distributions

03/18/2023

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by   Armine Bagyan, et al.

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