Strong law of large numbers for generalized operator means

Abstract

Sturm's strong law of large numbers in $\mathrm{CAT}\left(0\right)$ spaces and in the Thompson metric space of positive invertible operators is not only an important theoretical generalization of the classical strong law but also serves as a root-finding algorithm in the spirit of a proximal point method with splitting. It provides an easily computable stochastic approximation based on inductive means. The purpose of this paper is to extend Sturm's strong law and its deterministic counterpart, known as the “nodice” version, to unique solutions of nonlinear operator equations that generate exponentially contracting ODE flows in the Thompson metric. This includes a broad family of so-called generalized (Karcher) operator means introduced by Pálfia in 2016. The setting of the paper also covers the framework of order-preserving flows on Thompson metric spaces, as investigated by Gaubert and Qu in 2014, and provides a generally applicable resolvent theory for this setting.