Laplace approximations for sums of independent random vectors

Let X-i, i is an element of N, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let Phi be a mapping B --> R. Under a ce ntral limit theorem assumption, an asymptotic evaluation of Z(n) = E (exp ( n Phi (Sigma(i=1)(n) X-i/n))), up to a factor (1 + o(1)), has been gotten i n Bolthausen [1]. In this paper, we show that the same asymptotic evaluatio n can be gotten without the central limit theorem assumption.