# Subgeometric ergodicity and $\beta$-mixing

## TL;DR

This paper extends the connection between ergodicity and $eta$-mixing from geometric to subgeometric cases, providing new tools for analyzing Markov chains with slower convergence rates.

## Contribution

It demonstrates that subgeometric ergodicity implies $eta$-mixing with subgeometric decay, broadening the applicability of mixing results beyond geometric cases.

## Key findings

- Subgeometric ergodicity implies $eta$-mixing with subgeometric decay.
- New results for self-exciting threshold autoregressive models.
- Useful for analyzing Markov chains where geometric ergodicity fails.

## Abstract

It is well known that stationary geometrically ergodic Markov chains are $\beta$-mixing (absolutely regular) with geometrically decaying mixing coefficients. Furthermore, for initial distributions other than the stationary one, geometric ergodicity implies $\beta$-mixing under suitable moment assumptions. In this note we show that similar results hold also for subgeometrically ergodic Markov chains. In particular, for both stationary and other initial distributions, subgeometric ergodicity implies $\beta$-mixing with subgeometrically decaying mixing coefficients. Although this result is simple it should prove very useful in obtaining rates of mixing in situations where geometric ergodicity can not be established. To illustrate our results we derive new subgeometric ergodicity and $\beta$-mixing results for the self-exciting threshold autoregressive model.