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

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- by Xiongping Dai, Tingwen Huang and Yu Huang PDF
- Trans. Amer. Math. Soc.
**369**(2017), 5271-5292 Request permission

## 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.

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## Additional Information

**Xiongping Dai**- Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
- MR Author ID: 609395
- Email: xpdai@nju.edu.cn
**Tingwen Huang**- Affiliation: Science Program, Texas A$\&$M University at Qatar, P.O. Box 23874, Doha, Qatar
- Email: tingwen.huang@qatar.tamu.edu
**Yu Huang**- Affiliation: Department of Mathematics, Zhongshan (Sun Yat-Sen) University, Guangzhou 510275, People’s Republic of China
- MR Author ID: 197768
- Email: stshyu@mail.sysu.edu.cn
- Received by editor(s): August 19, 2015
- Published electronically: January 9, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 5271-5292 - MSC (2010): Primary 93E15, 37C75; Secondary 60J10, 93C30, 15A52, 93D20
- DOI: https://doi.org/10.1090/tran/6912
- MathSciNet review: 3646762