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2013 NO DICE THEOREM ON SYMMETRIC CONES
Sangho Kum, Hosoo Lee, Yongdo Lim
Taiwanese J. Math. 17(6): 1967-1982 (2013). DOI: 10.11650/tjm.17.2013.2944

Abstract

The monotonicity of the least squares mean on the Riemannian manifold of positive definite matrices, conjectured by Bhatia and Holbrook and one of key axiomatic properties of matrix geometric means, was recently established based on the Strong Law of Large Number [14, 4]. A natural question concerned with the S.L.L.N is so called the no dice conjecture. It is a problem to make a construction of deterministic sequences converging to the least squares mean without any probabilistic arguments. Very recently, Holbrook [7] gave an affirmative answer to the conjecture in the space of positive definite matrices. In this paper, inspired by the work of Holbrook [7] and the fact that the convex cone of positive definite matrices is a typical example of a symmetric cone (self-dual homogeneous convex cone), we establish the no dice theorem on general symmetric cones.

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Sangho Kum. Hosoo Lee. Yongdo Lim. "NO DICE THEOREM ON SYMMETRIC CONES." Taiwanese J. Math. 17 (6) 1967 - 1982, 2013. https://doi.org/10.11650/tjm.17.2013.2944

Information

Published: 2013
First available in Project Euclid: 10 July 2017

zbMATH: 1295.47008
MathSciNet: MR3141869
Digital Object Identifier: 10.11650/tjm.17.2013.2944

Subjects:
Primary: 15B48 , 17C50 , 47A64 , 53C20

Keywords: least squares mean , no dice theorem , symmetric cone

Rights: Copyright © 2013 The Mathematical Society of the Republic of China

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Vol.17 • No. 6 • 2013
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