Ergodicity of a Markov chain

2012-01-23T18:03:13Z

Hi,

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:

$(X_t)$ is irreducible.
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, $E_a[R_A]<\infty$ where $$ R_A := \inf ( t > 0 \mid X_t \in A ). $$
There exist $k,p>0$ such that for all $a,b \in A$, we have $$P (X_{t+k} = b \mid X_t = a) \geq p.$$

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.

Can some help show this claim or suggest why it's wrong?

Thanks Apus

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Ergodicity of a Markov chain