Asymptotic Expansions for the Laplace Approximations of Sums of Banach Space-valued Random Variables
نویسندگان
چکیده
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 intoR. An asymptotic evaluation of Zn = E(exp(nΦ( ∑n i=1 Xi/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.
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