High dimensional limit theorems and matrix decompositions on the Stiefel manifold

AbstractThe main purpose of this paper is to investigate high dimensional limiting behaviors, as m becomes infinite (m → ∞), of matrix statistics on the Stiefel manifold Vk, m, which consists of m × k (m ≥ k) matrices X such that X′X = Ik. The results extend those of Watson. Let X be a random matrix on Vk, m. We present a matrix decomposition of X as the sum of mutually orthogonal singular value decompositions of the projections PVX and PV⊥X, where V and V⊥ are each a subspace of Rm of dimension p and their orthogonal compliment, respectively (p ≥ k and m ≥ k + p). Based on this decomposition of X, the invariant measure on Vk, m is expressed as the product of the measures on the component subspaces. Some distributions related to these decompositions are obtained for some population distributions on Vk, m. We show the limiting normalities, as m → ∞, of some matrix statistics derived from the uniform distribution and the distributions having densities of the general forms f(PVX) and f(m1/2PVX) on Vk, m. Subsequently, applications of these high dimensional limit theorems are considered in some testing problems