Resumen de Complete asymptotic expansions and the high-dimensional Bingham distributions

Armine Bagyan, Donald Richards

• For, let X be a random vector having a Bingham distribution on, the unit sphere centered at the origin in, and let denote the symmetric matrix parameter of the distribution. Let be the normalizing constant of the distribution and let be the matrix of first-order partial derivatives of with respect to the entries of .We derive complete asymptotic expansions for and, as; these expansions are obtained subject to the growth condition that, the Frobenius norm of, satisfies for all d, where and. Consequently, we obtain for the covariance matrix of X an asymptotic expansion up to terms of arbitrary degree in. 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 and and we demonstrate the rapid convergence of the bounds to zero as r decreases.