Exact Asymptotics for some Probability Distributions on Compact Manifolds
紧流形上若干概率分布的精确渐近分析
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 $S^{d-1}$,…