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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 finite: integrable, that is, $$E_a[R_A] E_a[R_A]<\infty$ where $$ R_A := \inf ( t > 0 \mid X_t \in A )< \infty,$$ for all $a \in A$. $$
  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
show/hide this revision's text 3 fixed typo (thanks didier)

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 conditional on $X_0 \in A$= a$, the expected return time back to $$inf A$ is finite: $$E_a[R_A] := \inf ( t > 0 \mid X_t \in A : t > 0 ) $$ back to < \infty,$$ for all $a \in A$is finite.
  3. There exist $k,p>0$ and a state $\sigma\in A$ such that : if for all $X_t a,b \in A$, then the probability that we have $$P (X_{t+k} = b \sigma$ is at least $p$.mid X_t = a) \geq p.$$

In other words, I'm under the impression that (2) and (3) implies positive recurrence. This is because if I start at any $\sigma$, 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 $\sigma$ 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
show/hide this revision's text 2 fixed last line

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 conditional on $X_0 \in A$, the expected return time $$inf ( X_t \in A : t > 0 )$$ back to $A$ is finite.
  3. There exist $k,p>0$ and a state $\sigma\in A$ such that: if $X_t \in A$, then the probability that $X_{t+k} = \sigma$ is at least $p$.

In other words, I'm under the impression that (2) and (3) implies positive recurrence. This is because if I start at $\sigma$, then I take some finite time to return back to $A$. Then use the $k$ (finite no. of) steps to return back to $A$ \sigma$ to show postive recurrence. Whence (1) and (2) 2)+(3) imply ergodicity.

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

Thanks Apus
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