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Consider a Markov chain, for simplicity let us consider time discrete chains.

The problem

We consider a TDMC (Time Discrete Markov Chain) $(X_t)_{t \geq 0}$ with $X \in \mathcal{X}$ (having $\mathcal{X}$ as the set of the chain states) under the assumption that the chain is ergodic.

Without going deeper into the reason that make me ask such a strange question, I would like to make the chain non-ergodic.

The question

Is there a way to make an ergodic chain non-ergodic?

For example by adding states preserving the old ones?

What approaches in literature, does anybody know?

In particular

I would like to preserve the environment of the network, I mean that given certain initial states, the chain should have a limit behavior that is coherent. If for example I use an absorption state, as I was suggested in one answer, should I use one absorption state connected to all states or an absorption state for every node?

Thankyou

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Crosspost: stats.stackexchange.com/questions/26729/… –  cardinal Apr 19 '12 at 2:53
    
I voted for closing the one in stat –  Andry Apr 19 '12 at 4:39
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1 Answer 1

up vote 1 down vote accepted

Sure it's possible: just add one "absorbing state" with even a single incoming edge of nonzero probability, but no outgoing edges. Then, the chain is no longer ergodic because the original states are no longer positive recurrent.

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Thank you for the idea, please see my edit –  Andry Apr 19 '12 at 5:22
    
@andry: I edited in response. Even a single edge is enough. If the original chain was ergodic, entering the absorbing state is inevitable. –  Neil G Apr 19 '12 at 5:37
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