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

• 成果类型: Article
• 署名作者: Albeverio, S; Liang, S
• 署名单位: University of Bonn; Tohoku University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000001017
• 发表日期: 2005
• 页码: 300-336
• 关键词: independent random vectors; gaussian integrals; minimum points

摘要

Let X-i, i epsilon N, be i.i.d. B-valued randorn 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(nPhi(Sigma(i=1)(n) X-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 16 (2000) 221-238]. In this paper, a detailed asymptotic expansion of Z(n) as n --> infinity is given, valid to all orders, and with control on remainders. The results are new even in finite dimensions.