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.
Publié le : 1995-10-14
Classification:
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,
62E15,
62H11,
34E05,
58E05
@article{1176324313,
author = {Richards, Donald St. P.},
title = {Exact Asymptotics for some Probability Distributions on Compact Manifolds},
journal = {Ann. Statist.},
volume = {23},
number = {6},
year = {1995},
pages = { 1582-1586},
language = {en},
url = {http://dml.mathdoc.fr/item/1176324313}
}
Richards, Donald St. P. Exact Asymptotics for some Probability Distributions on Compact Manifolds. Ann. Statist., Tome 23 (1995) no. 6, pp. 1582-1586. http://gdmltest.u-ga.fr/item/1176324313/