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

2005/03/25 by Sergio Albeverio, Song Liang
Mathematics · #math.PR #msc:62E20 #msc:60F10 #msc:60B12.

paper · pdf · doi:10.1214/009117904000001017

published as Annals of Probability 2005, Vol. 33, No. 1, 300-336 · Published at http://dx.doi.org/10.1214/009117904000001017 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

arxiv created 2005/03/25 · arxiv updated 2009/12/01

Abstract

Let X_i, i∈ N, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let Φbe a smooth enough mapping from B into R. An asymptotic evaluation of Z_n=E(exp (nΦ(∑_i=1ⁿX_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→ ∞ is given, valid to all orders, and with control on remainders. The results are new even in finite dimensions.

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