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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].

MSC:

60B05 Probability measures on topological spaces
60F15 Strong limit theorems
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