Ergodicity of a Markov chain - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T04:13:21Z http://mathoverflow.net/feeds/question/86467 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86467/ergodicity-of-a-markov-chain Ergodicity of a Markov chain A Chuh 2012-01-23T18:03:13Z 2012-01-29T19:41:32Z <p>Hi,</p> <p>I'd appreciate some help on a Markov chain result I'm trying to show. I believe the following is sufficient for a continuous time Markov chain $(X_t)$ with a countable state space to be ergodic:</p> <ol> <li>$(X_t)$ is irreducible. </li> <li>There exists a finite subset $A$ of states such that, for all $a\in A$, conditional on $X_0 = a$, the expected return time back to $A$ is integrable, that is, $E_a[R_A]&lt;\infty$ where $$R_A := \inf ( t > 0 \mid X_t \in A ).$$</li> <li>There exist $k,p>0$ such that for all $a,b \in A$, we have $$P (X_{t+k} = b \mid X_t = a) \geq p.$$</li> </ol> <p>In other words, I'm under the impression that (2) and (3) implies positive recurrence. This is because if I start at any $a\in A$, then I take some finite time to return back to $A$. Then use the $k$ (finite no. of) steps to return back to $a$ to show postive recurrence. Whence (1) and (2)+(3) imply ergodicity.</p> <p>Can some help show this claim or suggest why it's wrong?</p> <p>Thanks Apus</p>