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

Sergio Albeverio·Song Liang
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Abstract

Let X_i, i\in 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(\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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Sergio Albeverio, Song Liang. 2005-03-25. Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables. https://doi.org/10.1214/009117904000001017

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