Infinite Excess Entropy Processes with Countable-State Generators

TL;DR

This paper introduces two novel examples of infinite excess entropy processes generated by countable-state hidden Markov models, expanding the understanding of such processes beyond previous uncountable-state constructions.

Contribution

It provides the first known constructions of infinite excess entropy processes with countable-state HMMs, including both ergodic and non-ergodic examples.

Findings

First example is non-ergodic

Second example is ergodic

Both are generated by countable-state HMMs

Abstract

We present two examples of finite-alphabet, infinite excess entropy processes generated by invariant hidden Markov models (HMMs) with countable state sets. The first, simpler example is not ergodic, but the second is. It appears these are the first constructions of processes of this type. Previous examples of infinite excess entropy processes over finite alphabets admit only invariant HMM presentations with uncountable state sets.