Higher Asymptotics of Laplace's Approximation

Details Higher Asymptotics of Laplace's Approximation Higher Asymptotics of Laplace's Approximation

2008-10-09

arxiv.org/abs/0810.1700v2

Mathematics

Classical Analysis and ODEs

8 pages; v2 significant expansion, several corollaries of main theorem added, minor corrections

Scientific paper

We present expressions for the coefficients which arise in asymptotic expansions of multiple integrals of Laplace type (the first term of which is known as Laplace's approximation) in terms of asymptotic series of the functions in the integrand. Our most general result assumes no smoothness of the functions of the integrand, but the expressions we obtain contain integrals which may be difficult to evaluate in practice. We then make additional assumptions which are sufficient to simplify these integrals, in some cases obtaining explicit formulae for the coefficients in the asymptotic expansions.

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