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February, 2018 The Order Properties and Karcher Barycenters of Probability Measures on the Open Convex Cone
Sejong Kim
Taiwanese J. Math. 22(1): 79-94 (February, 2018). DOI: 10.11650/tjm/8117

Abstract

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.

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Sejong Kim. "The Order Properties and Karcher Barycenters of Probability Measures on the Open Convex Cone." Taiwanese J. Math. 22 (1) 79 - 94, February, 2018. https://doi.org/10.11650/tjm/8117

Information

Received: 25 December 2016; Revised: 17 March 2017; Accepted: 21 May 2017; Published: February, 2018
First available in Project Euclid: 17 August 2017

zbMATH: 06965361
MathSciNet: MR3749356
Digital Object Identifier: 10.11650/tjm/8117

Subjects:
Primary: 47B65
Secondary: 15B48

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

Rights: Copyright © 2018 The Mathematical Society of the Republic of China

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Vol.22 • No. 1 • February, 2018
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