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


Authors: Xiongping Dai, Tingwen Huang and Yu Huang
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
Published electronically: January 9, 2017
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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

$\displaystyle \mathbb{P}\left (\Vert\xi _0(\omega )\dotsm \xi _{n-1}(\omega )\Vert\le C\lambda ^{n}\ \forall n\ge 1\right )=1;$    

and $ \boldsymbol {\xi }$ is called nonuniformly exponentially stable if there exist two random variables $ C(\omega )>0$ and $ 0<\lambda (\omega )<1$ such that

$\displaystyle \mathbb{P}\left (\Vert\xi _0(\omega )\dotsm \xi _{n-1}(\omega )\Vert\le C(\omega )\lambda (\omega )^{n}\ \forall n\ge 1\right )=1.$    

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
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
Email: stshyu@mail.sysu.edu.cn

DOI: https://doi.org/10.1090/tran/6912
Received by editor(s): August 19, 2015
Published electronically: January 9, 2017
Article copyright: © Copyright 2017 American Mathematical Society