Exact asymptotics of Laplace integrals on nonsmooth functions. (Q1889490)

The author determines the exact asymptotics (including estimates for the remainder term) of the Laplace type integral \[ I(\lambda)=\int_X g(x) \exp(\lambda S(x))\,\mu(dx) \quad \text{ as } \lambda \to \infty\,. \] Here \((X,B;\mu)\) is a finite measure space, and \(g\) and \(S\) are real valued measurable functions such that \[ \text{ess sup}_{x\in X} S(x) < \infty, \quad \text{ess sup}_{x\in X} | g(x)| < \infty, \quad \mu\{x\in X: g(x)=0\}=0\,. \] In particular, no assumptions on the topological structure of \(X\) are made, and no smoothness conditions on \(g\) and \(S\) are posed. Some concrete examples and an application to probability theory are given.