Ergodic Theorems for Dynamic Imprecise Probability Kinematics

Abstract

We formulate an ergodic theory for the (almost sure) limit $\mathcal{P}^\text{co}_{\tilde{\mathcal{E}}}$ of a sequence $(\mathcal{P}^\text{co}_{\mathcal{E}_n})$ of successive dynamic imprecise probability kinematics (DIPK, introduced in Caprio and Gong, 2021) updates of a set $\mathcal{P}^\text{co}_{\mathcal{E}_0}$ representing the initial beliefs of an agent. As a consequence, we formulate a strong law of large numbers.