Optimal tracking for dynamical systems

Kevin McGoff, Andrew Nobel

Published 2016-01-19Version 1

We study the limiting behavior of the average per-state cost when trajectories of a topological dynamical system are used to track a trajectory from an observed ergodic system. We establish a variational characterization of the limiting average cost in terms of dynamically invariant couplings, also known as joinings, of the two dynamical systems, and we show that the set of optimal joinings is convex and compact in the weak topology. Using these results, we establish a general convergence theorem for the limiting behavior of statistical inference procedures based on optimal tracking. The setting considered here is general enough to encompass traditional statistical problems with weakly dependent, real-valued observations. As applications of the general inference result, we consider the consistency of regression estimation under ergodic sampling and of system identification from quantized observations.