Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables
Sergio Albeverio and Song Liang
Source: Ann. Probab. Volume 33, Number 1 (2005), 300-336.
Abstract
Let Xi, i∈N, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let Φ be a smooth enough mapping from B into R. An asymptotic evaluation of Zn=E(exp(nΦ(∑i=1nXi/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 Zn as n→∞ is given, valid to all orders, and with control on remainders. The results are new even in finite dimensions.
Primary Subjects: 62E20, 60F10, 60B12
Keywords: Laplace approximation; asymptotic expansions; i.i.d. random vectors; Banach space-valued random variables
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1108141728
Digital Object Identifier: doi:10.1214/009117904000001017
Mathematical Reviews number (MathSciNet):
MR2118867
Zentralblatt MATH identifier:
1092.62024
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