Computing Gaussian and exponential integrals in ${\Bbb R}^n$

We consider expectations of the type $E\ \exp \left\{\sum_{i=1}^m \phi_i \right\}$, where $\phi_i: {\Bbb R}^n \longrightarrow {\Bbb C}$ are functions, each depending on a few coordinates of a point in ${\Bbb R}^n$, and the expectation is taken with respect to the standard Gaussian or symmetric exponential probability measures.

We prove sufficient conditions, in terms of the Lipschitz constants of $\phi_i$ and the combinatorics of their dependencies, for the integral to be non-zero, and, consequently, to be amenable to a computationally efficient approximation.

We discuss applications to computing volumes of bodies and statistics on integer points in polyhedra in ${\Bbb R}^n$.