Laplace approximations for sums of independent random vectors (Q5902833)

Let \(X_ i\), \(i\in {\mathbb{N}}\), be i.i.d. B-valued random variables where B is a real separable Banach space, and \(\Phi\) a mapping \(B\to {\mathbb{R}}\). Under some conditions an asymptotic evaluation of \(Z_ n=E(\exp (n\Phi (\sum^{n}_{i=1}X_ i/n)))\) is possible, up to a factor \((1+o(1))\). This also leads to a limit theorem for the appropriately normalized sums \(\sum^{n}_{i=1}X_ i\) under the law transformed by the density exp(n\(\Phi\) (\(\sum^{n}_{i=1}X_ i/n))/Z_ n\).