Let X i, 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 Z n = E(exp(nφ(∑ 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 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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