Exponential stability of matrix-valued Markov chains via nonignorable periodic data

Abstract:

Let $\boldsymbol {\xi }=\{\xi _n\}_{n\ge 0}$ be a Markov chain defined on a probability space $(\Omega ,\mathscr {F},\mathbb {P})$ valued in a discrete topological space $\boldsymbol {S}$ that consists of a finite number of real $d\times d$ matrices. As usual, $\boldsymbol {\xi }$ is called uniformly exponentially stable if there exist two constants $C>0$ and $0<\lambda <1$ such that \begin{gather*} \mathbb {P}\left (\|\xi _0(\omega )\dotsm \xi _{n-1}(\omega )\|\le C\lambda ^{n}\ \forall n\ge 1\right )=1; \end{gather*} and $\boldsymbol {\xi }$ is called nonuniformly exponentially stable if there exist two random variables $C(\omega )>0$ and $0<\lambda (\omega )<1$ such that \begin{gather*} \mathbb {P}\left (\|\xi _0(\omega )\dotsm \xi _{n-1}(\omega )\|\le C(\omega )\lambda (\omega )^{n}\ \forall n\ge 1\right )=1. \end{gather*} In this paper, we characterize the exponential stabilities of $\boldsymbol {\xi }$ via its nonignorable periodic data whenever $\boldsymbol {\xi }$ has a constant transition binary matrix. As an application, we construct a Lipschitz continuous $\mathrm {SL}(2,\mathbb {R})$-cocycle driven by a Markov chain with $2$-points state space, which is nonuniformly but not uniformly hyperbolic and which has constant Oseledeč splitting with respect to a canonical Markov measure.