Ergodicity of a Markov chain

Question

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:

1. $(X_t)$ is irreducible.
2. 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 $$ R_A := \inf ( t > 0 \mid X_t \in A ). $$
3. 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

Answer 1

Theorem 3.5.3: Let the intensity matrix $Q$ be irreducible. Then if some state is positive recurrent, every state is recurrent. Further there exists a stationary distribution $\lambda$.

-Some states (all in $A$) are positive recurrent (i.e. return time has finite expectation) which implies that all states are positive recurrent and that there exists a stationary distribution.

Please define what you mean by ergodic. For example there is a theorem which states that for a irreducible positive recurrent chain,

Theorem 3.8.1: $$\mathbb{P}\Big(\frac{1}{t} \int_0^t f(X_s)\,ds \rightarrow \bar{f} \text{ as } t \rightarrow\infty\Big)=1,$$ where $\bar{f} = \sum \lambda_j f_j$, and $\lambda$ is the stationary distribution.

But sometimes ergodic means that the limit of the probability exists.

Source: "Markov Chains" by Norris