Contractive barycentric maps and L1 ergodic theorems on the cone of positive definite matrices

We are concerned with contractive (with respect to the Wasserstein metric) barycenters of probability measures with bounded support on the convex cone of positive definite matrices equipped with the Thompson metric. Based on the important construction schemes of multivariate matrix means, namely the proximal average, and the Cartan mean (the least squares average) for the Cartan-Hadamard metric, we construct a one parameter family of contractive barycentric maps interpolating continuously and monotonically the harmonic, arithmetic and Cartan barycenters. We show that each contractive barycentric map is monotonic for the stochastic order induced by the cone and establish stochastic approximations and L1 ergodic theorems for the parameterized contractive barycenters.