Exact Asymptotics for some Probability Distributions on Compact Manifolds

Abstract

Let $M$ be a compact, smooth, orientable manifold without boundary, and let $f: M \rightarrow \mathbb{R}$ be a smooth function. Let $dm$ be a volume form on $M$ with total volume 1, and denote by $X$ the corresponding random variable. Using a theorem of Kirwan, we obtain necessary conditions under which the method of stationary phase returns an exact evaluation of the characteristic function of $f(X)$. As an application to the Langevin distribution on the sphere $S^{d-1}$, we deduce that the method of stationary phase provides an exact evaluation of the normalizing constant for that distribution when, and only when, $d$ is odd.