Wasserstein barycenters of probability measures

In this work we discuss Wasserstein barycenters of probability measures with support in R^d. In particular, we consider a probabilistic setting where the probability measures are considered to be random objects. Our main focus lies on Gaussian probability measures. We study the explicit form of the Wasserstein distance and the barycenter for these measures. We also consider a probability distribution on a set of Gaussian measures with commuting covariance matrices for which we prove the explicit form of the population barycenter and the consistency of the sample estimate. In addition, we proof the validity of a bootstrap procedure that allows to compute confidence sets for the population barycenter of this distribution.