Taiwanese Journal of Mathematics

The Order Properties and Karcher Barycenters of Probability Measures on the Open Convex Cone

Sejong Kim

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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.

Article information

Taiwanese J. Math., Volume 22, Number 1 (2018), 79-94.

Received: 25 December 2016
Revised: 17 March 2017
Accepted: 21 May 2017
First available in Project Euclid: 17 August 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B65: Positive operators and order-bounded operators
Secondary: 15B48: Positive matrices and their generalizations; cones of matrices

Loewner order stochastic order Wasserstein distance max-flow and min-cut theorem unitarily invariant norm Karcher barycenter


Kim, Sejong. The Order Properties and Karcher Barycenters of Probability Measures on the Open Convex Cone. Taiwanese J. Math. 22 (2018), no. 1, 79--94. doi:10.11650/tjm/8117. https://projecteuclid.org/euclid.twjm/1502935230

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