Geodesic
A combinatorial approach to nearly uncoupled Markov chains. I: Reversible Markov chains
Electronic transactions on numerical analysis, Tome 40 (2013), pp. 120-147
A Markov chain is a sequence of random variables $x_{0} , x_{1} , \ldots$ that take values in a state space $\mathcal{S}$. A set $\mathcal{E} \subseteq \mathcal{S}$ is referred to as an almost invariant aggregate if transitions from $x_{t}$ to $x_{t+1}$ where $x_{t} \in \mathcal{E}$ and $x_{t+1} \notin \mathcal{E}$ are exceedingly rare. A Markov chain is referred to as nearly uncoupled if there are two or more disjoint almost invariant aggregates contained in its state space. Nearly uncoupled Markov chains are characterised by long periods of relatively constant behaviour punctuated by sudden, extreme changes. We present an algorithm for producing almost invariant aggregates of a nearly uncoupled reversible Markov chain. This algorithm utilises the stochastic complement to iteratively reduce the order of the given state space.
Classification : 15A18, 15A51, 60J10, 60J20, 65F15
Keywords: nearly uncoupled Markov chain, reversible Markov chain, stochastic complement, stochastic matrix 
@article{ETNA_2013__40__a19,
     author = {Tifenbach,  Ryan M.},
     title = {A combinatorial approach to nearly uncoupled {Markov} chains. {I:} {Reversible} {Markov} chains},
     journal = {Electronic transactions on numerical analysis},
     pages = {120--147},
     year = {2013},
     volume = {40},
     zbl = {1288.65011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2013__40__a19/}
}
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Tifenbach,  Ryan M. A combinatorial approach to nearly uncoupled Markov chains. I: Reversible Markov chains. Electronic transactions on numerical analysis, Tome 40 (2013), pp. 120-147. http://geodesic.mathdoc.fr/item/ETNA_2013__40__a19/