The Generalized Entropy Ergodic Theorem for Nonhomogeneous Markov Chains

TL;DR

This paper extends the classical entropy ergodic theorem to nonhomogeneous Markov chains, establishing almost-everywhere and
L_1 convergence results for chains with finite state spaces.

Contribution

It introduces a generalized entropy ergodic theorem applicable to nonhomogeneous Markov chains, broadening the scope of classical ergodic results.

Findings

Proves almost-everywhere convergence of entropy for nonhomogeneous chains

Establishes
L_1 convergence under generalized conditions

Extends classical results to more general Markov processes

Abstract

Let $(\xi_n)_{n=0}^\infty$ be a nonhomogeneous Markov chain taking values from finite state-space of $\mathbf{X}=\{1,2,\ldots,b\}$. In this paper, we will study the generalized entropy ergodic theorem with almost-everywhere and $\mathcal{L}_1$ convergence for nonhomogeneous Markov chains, which generalizes the corresponding classical results for the Markov chains.