A game-theoretic ergodic theorem for imprecise Markov chains
Invited plenary lecture at the Fifth Workshop on Game-Theoretic Probability and Related Topics (GTP2014), held in Guanajuato, Mexico, 10-13 November 2014.
Based on research I’ve done in the past year with Jasper De Bock and Stavros Lopatatzidis. I had been trying to prove an ergodic theorem in an imprecise probabilities context ever since I started work in the field. These things take time and patience, especially in imprecise probabilities, because many of the tools present in the precise counterpart are simply not available yet. Or take surprisingly different forms.
Here’s the abstract: We prove a game-theoretic version of the strong law of large numbers for submartingale differences, and use this to derive a pointwise ergodic theorem for discrete-time Markov chains with finite state sets, when the transition probabilities are imprecise, in the sense that they are only known to belong to some convex closed set of probability measures.