Persistence exponents in Markov chains

@article{Aurzada2021PersistenceEI,
  title={Persistence exponents in Markov chains},
  author={Frank Aurzada and Sumit Mukherjee and Ofer Zeitouni},
  journal={Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques},
  year={2021}
}
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 $\{X_i\}_{i\geq 0}$ in a Polish space, where $S$ is a Borel measurable set and $\mu$ is the initial distribution. Focusing on the case of AR($p$) and MA($q$) processes with $p,q\in N$ and continuous innovation distribution, we study the existence of $\lambda$ and its continuity in the parameters, for $S… 
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