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

TL;DR

This paper introduces a parsing framework for stationary processes based on stopping times, providing new insights into pointwise convergence theorems in ergodic theory through asymptotic analysis.

Contribution

It develops a novel parsing approach using stopping times that links weak convergence results to classical pointwise ergodic theorems.

Findings

Derives pointwise ergodic theorem from convergence in probability.

Establishes Shannon-McMillan-Breiman theorem via parsing methods.

Provides a unified framework connecting weak and almost sure convergence.

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.