Laplace approximations for sums of independent random vectors

Let X i , i∈ℕ, be i.i.d. B-valued random variables where B is a real separable Banach space, and Φ a mapping B→ℝ. Under some conditions an asymptotic evaluation of Z n =E(exp(nΦ(∑ i=1 n X i /n))) is possible, up to a factor (1+o(1)). This also leads to a limit theorem for the appropriately normalized sums ∑ i=1 n X i under the law transformed by the density exp(nΦ (∑ i=1 n X i /n))/Z n