Convergence theorems for barycentric maps

TL;DR

This paper develops a comprehensive theory of conditional expectations and convergence theorems for random variables in metric spaces with barycentric maps, extending classical probabilistic results to more general settings.

Contribution

It introduces a new framework for conditional expectations in metric spaces with barycentric maps and establishes convergence theorems for martingales, ergodic theorems, and large deviations.

Findings

Established convergence theorems for $eta$-martingales.

Proved a Birkhoff ergodic theorem for $eta$-values.

Derived large deviation principles for empirical measures.

Abstract

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 $\beta$, and then give convergence theorems for martingales of $\beta$-conditional expectations. We give the Birkhoff ergodic theorem for $\beta$-values of ergodic empirical measures and provide a description of the ergodic limit function in terms of the $\beta$-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 $\beta$-values of i.i.d. empirical measures is obtained by applying the Sanov large deviation principle.