Asymptotic Approximations for Probability Integrals

This is a self-contained introduction to the asymptotic approximation of multivariate integrals for both mathematicians and applied scientists. A collection of results of the Laplace methods is given. Such methods are useful in reliability, statistics, theoretical physics and information theory. An important special case is the approximation of multidimensional normal integrals. Here the relation between the differential geometry of the boundary of the integration domain and the asymptotic probability content is derived.