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The order properties and Karcher barycenters of probability measures on the open convex cone. (English) Zbl 1505.60007

Summary: We study the probability measures on the open convex cone of positive definite operators equipped with the Loewner ordering. We show that two crucial push-forward measures derived by the congruence transformation and power map preserve the stochastic order for probability measures. By the continuity of two push-forward measures with respect to the Wasserstein distance, we verify several interesting properties of the Karcher barycenter for probability measures with finite first moment such as the invariant properties and the inequality for unitarily invariant norms. Moreover, the characterization for the stochastic order of uniformly distributed probability measures has been shown.

MSC:

60B05 Probability measures on topological spaces
47B65 Positive linear operators and order-bounded operators
15B48 Positive matrices and their generalizations; cones of matrices
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