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

Abstract

Let ξ = { ξ n } n ≥ 0 \boldsymbol {\xi }=\{\xi _n\}_{n\ge 0} be a Markov chain defined on a probability space ( Ω , F , P ) (\Omega ,\mathscr {F},\mathbb {P}) valued in a discrete topological space S \boldsymbol {S} that consists of a finite number of real d × d d\times d matrices. As usual, ξ \boldsymbol {\xi } is called uniformly exponentially stable if there exist two constants C > 0 C>0 and 0 > λ > 1 0>\lambda >1 such that P ( ‖ ξ 0 ( ω ) ⋯ ξ n − 1 ( ω ) ‖ ≤ C λ n ∀ n ≥ 1 ) = 1 ; \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 ξ \boldsymbol {\xi } is called nonuniformly exponentially stable if there exist two random variables C ( ω ) > 0 C(\omega )>0 and 0 > λ ( ω ) > 1 0>\lambda (\omega )>1 such that P ( ‖ ξ 0 ( ω ) ⋯ ξ n − 1 ( ω ) ‖ ≤ C ( ω ) λ ( ω ) n ∀ n ≥ 1 ) = 1. \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 ξ \boldsymbol {\xi } via its nonignorable periodic data whenever ξ \boldsymbol {\xi } has a constant transition binary matrix. As an application, we construct a Lipschitz continuous S L ( 2 , R ) \mathrm {SL}(2,\mathbb {R}) -cocycle driven by a Markov chain with 2 2 -points state space, which is nonuniformly but not uniformly hyperbolic and which has constant Oseledeč splitting with respect to a canonical Markov measure.