Théoràme ergodique ponctuel et lois fortes des grands nombres pour des points aléatoires d'un espace métrique à courbure négative

Abstract

Let $M$ be a complete separable metric space with negative curvature as defined by Herer. Using Herer’s definition of the mathematical expectation of a random point of $M$, we extend to sequences of random points of $M$ a pointwise ergodic theorem and strong laws of large numbers (SLLN), known in the case where $M$ is a separable Banach space (SLLN of Etemadi, of Beck and Giesy and of Cuesta and Matrán). The convergence results obtained here are stated for the Hausdorff topology or the Wijsman topology in the space of closed subsets of $M$.