# Markov chains with exponential return times are finitary

## TL;DR

This paper proves that ergodic Markov chains with exponential tail return times can be represented as finitary factors of i.i.d. processes, extending to renewal processes with exponential tail jumps.

## Contribution

It establishes that Markov chains with exponential return times are finitary factors of i.i.d. processes, including renewal processes with exponential tail jumps, broadening the understanding of such stochastic processes.

## Key findings

- Markov chains with exponential return times are finitary factors of i.i.d. processes.
- Stationary renewal processes with exponential tail jumps are also finitary factors.
- The results apply to chains not supported on proper subgroups of integers.

## Abstract

Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an i.i.d. process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of $\mathbb{Z}$ is a finitary factor of an i.i.d. process.