EXACT ASYMPTOTICS FOR SOME PROBABILITY-DISTRIBUTIONS ON COMPACT MANIFOLDS

Let M be a compact, smooth, orientable manifold without boundary, and let f: M --> 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. Us ing a theorem of Kirwan, we obtain necessary conditions under which th e method of stationary phase returns an exact evaluation of the charac teristic function of f(X). As an application to the Langevin distribut ion on the sphere S-d-1, We deduce that the method of stationary phase provides an exact evaluation of the normalizing constant for that dis tribution when, and only when, d is odd.