Sets of Probability Measures and Convex Combination Spaces

Abstract

The Wasserstein distances between probability distributions are an important tool in modern probability theory which has been generalized to sets of probability distributions. We will show that the (generalized) L1-Wasserstein metric, with the operations of convolution and rescaling, fits in the abstract framework of convex combination spaces: nonlinear metric spaces preserving some of the nice properties of a normed space but accomodating other unusual behaviours. For instance, unlike in a linear space, a singleton {𝑃} is typically not convex (it is so only if 𝑃 is degenerate). Also, some theorems for convex combination spaces are applied to this setting.