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Let X_i, i\in N, be i.i.d. B-valued random variables, where B is a realseparable Banach space. Let \Phi be a smooth enough mapping from B into R. Anasymptotic evaluation of Z_n=E(\exp (n\Phi (\sum_{i=1}^nX_i/n))), up to afactor (1+o(1)), has been gotten in Bolthausen [Probab. Theory Related Fields72 (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. Theresults are new even in finite dimensions.

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