Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?


share|cite|improve this question
Crosspost:… – cardinal Apr 19 '12 at 2:53
I voted for closing the one in stat – Andry Apr 19 '12 at 4:39
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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.