## Laplace Approximations for Sums of Independent Random Vectors -- The Degenerate Case --

J. Math. Sci. Univ. Tokyo
Vol. 7 (2000), No. 2, Page 195--220.

Liang, Song
Laplace Approximations for Sums of Independent Random Vectors -- The Degenerate Case --
Let $X_i, i \in {\bf N}$, be {\it i.i.d.} $B$-valued random variables, where $B$ is a real separable Banach space. Let $Î¦: B \to {\bf R}$ be a mapping. The problem is to give an asymptotic evaluation of $Z_n = E \left( \exp \left( n Î¦ (\sum_{i=1}^n X_i / n ) \right) \right)$, up to a factor $(1 + o(1))$. Bolthausen \cite{Bolthausen} studied this problem in the case that there is a unique point maximizing $Î¦ - h$, where $h$ is the so-called entropy function, and the curvature at the maximum is nonvanishing, (these two will be called as {\it nondegenerate assumptions}), with some central limit theorem assumption. Kusuoka-Liang \cite{K-L} studied the same problem, and succeeded in eliminating 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 nondegenerate assumptions.