Laplace Approximations for Sums of Independent Random Vectors, Probab

Abstract. Let Xi, i ∈ N, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let Φ: B → R be a mapping. The problem is to give an asymptotic evaluation of Zn = E (exp (nΦ( ∑n i=1 Xi/n))), up to a factor (1 + o(1)). Bolthausen [1] studied this problem in the case that there is a unique point maximiz-ing Φ−h, where h is the so-called entropy function, and the curvature at the maximum is nonvanishing, (these two will be called as nonde-generate assumptions), with some central limit theorem assumption. Kusuoka-Liang [5] studied the same problem, and succeeded in elim-inating the central limit theorem assumption, but the nondegenerate assumptions are still left. In this paper, we study the same problem not assuming the central limit theorem assumption and the nondegen-erate assumptions. 1