Abstract Sturm's strong law of large numbers in CAT ( 0 ) spaces has been an influential tool to study the geometric mean or also called Karcher barycenter of positive definite matrices. It provides an easily computable stochastic approximation based on inductive means. Convergence of a deterministic version of this approximation has been proved by Holbrook, providing his “nodice” theorem for the Karcher mean of positive definite matrices. The Karcher mean has also been extended to the infinite dimensional case of positive operators on a Hilbert space by Lawson-Lim and then to probability measures with bounded support by the second author, however the CAT ( 0 ) property of the space is lost and one defines the mean as the unique solution of a nonlinear operator equation on a convex Banach-Finsler manifold. The formulations of Sturm's strong law of large numbers and Holbrook's “nodice” approximation are natural and both conjectured to converge, however all previous techniques of their proofs break down, due to the Banach-Finsler nature of the space. In this paper we prove both conjectures by establishing the most general L 1 -form of Sturm's strong law of large numbers and Holbrook's “nodice” theorem in the operator norm by developing a stochastic discrete-time resolvent flow for the Karcher barycenter using its Wasserstein contraction property.

中文翻译：

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L1-Karcher 均值的强大数定律

摘要 斯图姆在 CAT ( 0 ) 空间中的强大数定律一直是研究正定矩阵的几何均值或称为卡歇重心的有影响力的工具。它提供了一种基于归纳方法的易于计算的随机近似。Holbrook 证明了这种近似的确定性版本的收敛性，提供了他的正定矩阵 Karcher 均值的“节点”定理。Karcher 均值也被 Lawson-Lim 扩展到希尔伯特空间上正算子的无限维情况，然后被第二作者扩展到有界支持的概率测度，但是空间的 CAT ( 0 ) 属性丢失了，并且将均值定义为凸 Banach-Finsler 流形上非线性算子方程的唯一解。Sturm'的配方 强大的大数定律和 Holbrook 的“节点”近似是自然的，并且都被推测收敛，但是由于空间的 Banach-Finsler 性质，所有先前的证明技术都失效了。在本文中，我们通过在算子范数中建立 Sturm 强大数定律的最一般 L 1 形式和 Holbrook 的“节点”定理，通过使用其 Wasserstein 收缩开发 Karcher 重心的随机离散时间解析流来证明这两个猜想财产。