2018/05/22 by Fumio Hiai, Hiai, Fumio, Yongdo Lim +1
Mathematics · #47H25 #60B05 #60G48 #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #math.FA #math.PR #msc:47H25 #msc:60B05 #msc:60G48
paper · pdf · doi:10.48550/arxiv.1805.08558
37 pages
arxiv created 2018/05/22 · arxiv updated 2018/05/23
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 derivation property of β-values of i.i.d. empirical measures is obtained by applying the Sanov large deviation principle.