Vitali variation error bounds for expected value functions

Abstract

We derive error bounds for two-dimensional expected value functions that depend on the Vitali variation of the joint probability density function of the corresponding random vector. Contrary to bounds from the literature, our bounds are not restricted to underlying functions that are one-dimensional and periodic. In our proof, we first derive the bounds in a discrete setting, which requires us to characterize the extreme points of the set of all matrices that have zero-sum rows and columns and have an $L_1$-norm bounded by one. This result may be of independent interest.