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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:459

Article

Lim, Yongdo¹ 

[1] Sungkyunkwan Univ, Dept Math, Suwon 440746, South Korea

关键词: Positive definite matrix;    Thompson metric;    Cartan mean;    Wasserstein distance;    Contractive barycenter;    L-1 ergodic theorem;   

DOI  :  10.1016/j.jmaa.2017.10.055

来源: Elsevier

【 摘 要 】

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 L-1 ergodic theorems for the parameterized contractive barycenters. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free