### Laplace's Method Revisited: Weak Convergence of Probability Measures

Chii-Ruey Hwang
Source: Ann. Probab. Volume 8, Number 6 (1980), 1177-1182.

#### Abstract

Let $Q$ be a fixed probability on the Borel $\sigma$-field in $R^n$ and $H$ be an energy function continuous in $R^n$. A set $N$ is related to $H$ by $N = \{x \mid\inf_yH(y) = H(x)\}$. Laplace's method, which is interpreted as weak convergence of probabilities, is used to introduce a probability $P$ on $N$. The general properties of $P$ are studied. When $N$ is a union of smooth compact manifolds and $H$ satisfies some smooth conditions, $P$ can be written in terms of the intrinsic measures on the highest dimensional mainfolds in $N$.

First Page:
Primary Subjects: 60B10
Secondary Subjects: 58C99
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.aop/1176994579
Digital Object Identifier: doi:10.1214/aop/1176994579
Mathematical Reviews number (MathSciNet): MR602391
Zentralblatt MATH identifier: 0452.60007