# Persistence exponents in Markov chains

## TL;DR

This paper establishes the existence and properties of persistence exponents for certain Markov chains, including AR and MA processes, providing new explicit calculations and analyzing their dependence on process parameters.

## Contribution

It proves the existence of persistence exponents for a broad class of Markov chains and studies their continuity and monotonicity in AR and MA models with explicit examples.

## Key findings

- Existence of persistence exponents for Markov chains in Polish spaces.
- Continuity of the persistence exponent in AR and MA process parameters.
- Explicit calculation of new persistence exponents in specific models.

## Abstract

We prove the existence of the persistence exponent $$\log\lambda:=\lim_{n\to\infty}\frac{1}{n}\log \mathbb{P}_\mu(X_0\in S,\ldots,X_n\in S)$$ for a class of time homogeneous Markov chains $\{X_i\}_{i\geq 0}$ taking values in a Polish space, where $S$ is a Borel measurable set and $\mu$ is an initial distribution. Focusing on the case of AR($p$) and MA($q$) processes with $p,q\in \mathbb{N}$ and continuous innovation distribution, we study the existence of $\lambda$ and its continuity in the parameters of the AR and MA processes, respectively, for $S=\mathbb{R}_{\geq 0}$. For AR processes with log-concave innovation distribution, we prove the strict monotonicity of $\lambda$. Finally, we compute new explicit exponents in several concrete examples.