2007, 17(4): 731-738. doi: 10.3934/dcds.2007.17.731

Stochastic matrix-valued cocycles and non-homogeneous Markov chains

1. 

Department of Applied Mathematics, University of Crete, GR-714 09 Heraklion, Crete, Greece

Received  March 2006 Revised  August 2006 Published  January 2007

We prove weak ergodicity theorems for non - homogeneous Markov chains $\{X_\nu\}_{\nu\geq 0}$ taking values in a finite state space $S=\{1,\cdots,n\}$ for which the family of transition matrices $\{g(x)\}_{x\in X}$ is generated from some underlying topological or measurable dynamical system $f:X\to X$. Using the projective metric of Hilbert on $\mathcal{S}=\{(x_1,\cdots,x_n)\in\mathbb R ^n : x_i\geq 0, x_1+\cdots+x_n=1\}$, the space of distributions, we form the skew-product $T:X\times\mathcal{S}\to X\times\mathcal{S}$ defined by $T(x,p)=(f(x),g(x)p)$ and show that, for continuous $g$ positive on some set, weak ergodicity for such processes is a result of the existence of a map $\gamma:X\to\mathcal{S}$ whose graph is attracting and invariant under $T$. Some results on random compositions of non-expansive maps are obtained on the way.
Citation: Demetris Hadjiloucas. Stochastic matrix-valued cocycles and non-homogeneous Markov chains. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 731-738. doi: 10.3934/dcds.2007.17.731
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