Asymptotic expansions for distributions of the large sample matrix resultant and related statistics on the Stiefel manifold

Let Vk,m denote the Stiefel manifold which consists of m - k(m >= k) matrices X such that X'X = Ik. Let X1,..., Xn be a random sample of size n from the matrix Langevin (or von Mises-Fisher) distribution on Vk,m, which has the density proportional to exp(tr F'X), with F an m - k matrix, and let Z = (m/n)1/2 [Sigma]j = 1n Xj. The exact expression of the distribution of Z in an integral form is intractable. In this paper, we derive asymptotic expansions, for large n and up to the order of n-3, for the distributions of Z, Z'Z, and related statistics in connection with testing problems on F, under the hypothesis of uniformity (F = 0) and local alternative hypotheses. In the derivation, we utilize zonal and invariant polynomials in matrix arguments and Hermite and Laguerre polynomials in one-dimensional variable and matrix argument.