Multivariate Laplace’s approximation with estimated error and application to limit theorems

摘要

In this paper we prove a Multivariate Laplace approximation with estimatederror for two cases of maximum, in the interior of the domain and on theboundary. Additionally, the function in the exponent and its point of maximumdepends on the integrals asymptotic parameter. As an application, we prove Lawof Large Numbers and the Central Limit Theorem for the random vector related tothe Laplace Integral. The second result gives different limiting distributionfor two cases of maximum in the Laplace integral. When it is in the interior ofthe domain it is Gaussian distribution and on the boundary it is exponential inone direction and Gaussian in other directions.