Parsings of Stationary Processes, Stopping Times and the Fundamental Pointwise Convergence Theorems of Ergodic Theory

Abstract: The idea of a parsing of a stationary process according to a collection of words is introduced, and the basic framework required for the asymptotic analysis of these parsings is presented. We demonstrate how the pointwise ergodic theorem and the Shannon-McMillan-Breiman theorem can be deduced from their respective weaker convergence in probability versions combined with our observations regarding parsings, where the parsings are done according to collections that originate in stopping times tailored for that purpose.