Is it possible to obtain Harris recurrent Markov chain from Ergodic chain (in Birkhoff sense) under certain assumption? That is, suppose we know a Markov chain is ergodic (in Birkhoff sense); is it possible to make it Harris recurrent under certain assumption?
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No - because the definition of Harris recurrence contains an absolute continuity condition on transition probabilities. For an extreme counterexample just take any ergodic measure preserving transformation and consider it as a deterministic Markov chain.
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$\begingroup$ If we restrict the ergodic chain, for example with compact state space, then is it possible to make it Harris recurrent? $\endgroup$ Commented Jun 3, 2015 at 18:37
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