Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables

Details Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables

2005-03-25

arxiv.org/abs/math/0503601v1

Annals of Probability 2005, Vol. 33, No. 1, 300-336

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117904000001017 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117904000001017

Let X_i, i\in N, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let \Phi be a smooth enough mapping from B into R. An asymptotic evaluation of Z_n=E(\exp (n\Phi (\sum_{i=1}^nX_i/n))), up to a factor (1+o(1)), has been gotten in Bolthausen [Probab. Theory Related Fields 72 (1986) 305-318] and Kusuoka and Liang [Probab. Theory Related Fields 116 (2000) 221-238]. In this paper, a detailed asymptotic expansion of Z_n as n\to \infty is given, valid to all orders, and with control on remainders. The results are new even in finite dimensions.

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