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October, 1995 Exact Asymptotics for some Probability Distributions on Compact Manifolds
Donald St. P. Richards
Ann. Statist. 23(5): 1582-1586 (October, 1995). DOI: 10.1214/aos/1176324313


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.


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Donald St. P. Richards. "Exact Asymptotics for some Probability Distributions on Compact Manifolds." Ann. Statist. 23 (5) 1582 - 1586, October, 1995.


Published: October, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0848.60020
MathSciNet: MR1370297
Digital Object Identifier: 10.1214/aos/1176324313

Primary: 62E15
Secondary: 34E05 , 58E05 , 62H11

Keywords: asymptotic expansion , Betti number , exponential model , Fisher distribution , hypergeometric function of matrix argument , Langevin distribution , method of stationary phase , Morse function , Morse inequalities , perfect Morse function , saddlepoint approximation

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 5 • October, 1995
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