# Subgeometrically ergodic autoregressions

## TL;DR

This paper explores how subgeometric ergodicity can be used to analyze the stationarity and mixing properties of higher-order nonlinear autoregressions, especially those resembling unit root processes, with slower convergence rates.

## Contribution

It extends the concept of subgeometric ergodicity from first-order to higher-order nonlinear autoregressions, providing new conditions for their ergodic behavior.

## Key findings

- Higher-order nonlinear autoregressions are subgeometrically ergodic under certain conditions.
- These models exhibit stationarity and β-mixing with subgeometrically decaying coefficients.
- The results generalize previous first-order cases to more complex autoregressive models.

## Abstract

In this paper we discuss how the notion of subgeometric ergodicity in Markov chain theory can be exploited to study stationarity and ergodicity of nonlinear time series models. Subgeometric ergodicity means that the transition probability measures converge to the stationary measure at a rate slower than geometric. Specifically, we consider suitably defined higher-order nonlinear autoregressions that behave similarly to a unit root process for large values of the observed series but we place almost no restrictions on their dynamics for moderate values of the observed series. Results on the subgeometric ergodicity of nonlinear autoregressions have previously appeared only in the first-order case. We provide an extension to the higher-order case and show that the autoregressions we consider are, under appropriate conditions, subgeometrically ergodic. As useful implications we also obtain stationarity and $\beta$-mixing with subgeometrically decaying mixing coefficients.