Persistence exponents in Markov chains

Abstract

We prove the existence of the persistence exponent $$\log \lambda :=\underset{n\to \infty }{\lim }\frac{1}{n}\log {\mathbb{P}}_{\mu }(X_0\in S,\dots ,X_n\in S)$$ for a class of time homogeneous Markov chains $\{X_i{\}}_{i\ge 0}$ 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 $S={\mathbb{R}}_{\ge 0}$. 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.

arXiv commentMinor changes. To appear in AIHP