Process Metrics and the Ergodic Decomposition

Given two probability measures, say p and m, on a common probability space, how different or distant from each other are they? Similarly, given two random processes with distributions p and m, how distant are the processes from each other and what impact does such a distance have on their respective ergodic properties? The goal of this final chapter is to develop two quite distinct notions of the distance d(p,m) between measures or processes and to use these ideas to delve further into the ergodic properties of processes and the ergodic decomposition. One metric, the distributional distance, measures how well the probabilities of certain important events match up for the two probability measures, and hence this metric need not have any relation to any underlying metric on the original sample space. In other words, the metric makes sense even when we are not putting probability measures on metric spaces. The second metric, the ρ̄-distance (rho-bar distance) depends very strongly on a metric on the output space of the process and measures distance not by how different probabilities are, but by how well one process can be made to approximate another. The second metric is primarily useful in applications in information theory and statistics.