Convergence theorems for barycentric maps

We first develop a theory of conditional expectations for random variables with values in a complete metric space [Formula: see text] equipped with a contractive barycentric map [Formula: see text], and then give convergence theorems for martingales of [Formula: see text]-conditional expectations. We give the Birkhoff ergodic theorem for [Formula: see text]-values of ergodic empirical measures and provide a description of the ergodic limit function in terms of the [Formula: see text]-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 [Formula: see text]. Finally, the large deviation property of [Formula: see text]-values of i.i.d. empirical measures is obtained by applying the Sanov large deviation principle.

Introduction to Large Deviation Theory

This chapter provides an introduction to large deviation theory. It begins with an overview of the motivatio n for the problem under study, focusing on probability distributions and how to construct an empirical distribution. It then considers the notion of a lower semi-continuous function and that of a lower semi-continuous relaxation before discussing the large deviation property for i.i.d. samples. In particular, it describes Sanov's theorem for a finite alphabet and proceeds by analyzing large deviation property for Markov chains, taking into account stationary distributions, entropy and relative entropy rates, the rate function for doubleton frequencies, and the rate function for singleton frequencies.