Convergence theorems for barycentric maps

We first develop a theory of conditional expectations for random variables with values in a complete metric space M equipped with a contractive barycentric map β, and then give convergence theorems for martingales of β-conditional expectations. We give the Birkhoff ergodic theorem for β-values of ergodic empirical measures and provide a description of the ergodic limit function in terms of the β-conditional expectation. Moreover, we prove the continuity property of the ergodic limit function by finding a complete metric between contractive barycentric maps on the Wasserstein space of Borel probability measures on M. Finally, the large deviation property of β-values of i.i.d. empirical measures is obtained by applying the Sanov large deviation principle.