Es-Sahib, Aziz; Heinich, Henri Canonical barycenter for a negatively curved metric space. (Barycentre canonique pour un espace métrique à courbure négative.) (French) Zbl 0952.60010 Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXIII. Berlin: Springer. Lect. Notes Math. 1709, 355-370 (1999). Consider a random variable \(X\) on a metric space \(M\); in the literature, the barycenter of \(X\) is often defined as a subset of \(M\); in particular, different definitions are due to Doss, Herer, and Emery-Mokobodzki. The aim of this work is to derive from each of these definitions another barycenter which will be a point of \(M\). Of course, this can be done only under some assumptions on \(M\) (convexity, negative curvature). First, the authors define the barycenter of a finite number of points; then they deduce the general case by taking the limit with a reverse martingale argument. They also verify that their barycenter satisfies the strong law of large numbers and the ergodic theorem.For the entire collection see [Zbl 0924.00016]. Reviewer: J.Picard (Aubière) Cited in 1 ReviewCited in 22 Documents MSC: 60B05 Probability measures on topological spaces 60F15 Strong limit theorems Keywords:probability measure on metric spaces; barycenters; strong law of large numbers; ergodic theorem PDFBibTeX XMLCite \textit{A. Es-Sahib} and \textit{H. Heinich}, Lect. Notes Math. 1709, 355--370 (1999; Zbl 0952.60010) Full Text: Numdam EuDML