  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.
