Computing Gaussian and exponential integrals in ${\Bbb R}^n$
2026-06-22 • Data Structures and Algorithms
Data Structures and Algorithms
AI summaryⓘ
The authors study a certain kind of average, called an expectation, involving sums of special functions that each look at just a few parts of a bigger input. They focus on cases where inputs follow either a normal (Gaussian) or symmetric exponential distribution. They find conditions under which this average stays away from zero, making it easier to calculate effectively. Their results help in estimating volumes of shapes and counting points with integer coordinates inside multi-dimensional polyhedra.
ExpectationGaussian measureSymmetric exponential distributionLipschitz constantCombinatoricsIntegral approximationVolume computationPolyhedraInteger pointsHigh-dimensional geometry
Authors
Alexander Barvinok
Abstract
We consider expectations of the type $E\ \exp \left\{\sum_{i=1}^m φ_i \right\}$, where $φ_i: {\Bbb R}^n \longrightarrow {\Bbb C}$ 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 $φ_i$ and the combinatorics of their dependencies, for the integral to be separated from 0, 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 ${\Bbb R}^n$.