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Suppose that you have a Markov chain with state space $E$ containing $0$. Assume that $$ p_{00}^{(2n)}=\binom{2n}{n}\left(\frac{1}{2}\right)^{2n-1}~~~\text{ and }~~~p_{00}^{(2n-1)}=0~~~\text{ for }n\in\mathbb{N}. $$ Is $0$ transient, positive recurrent or null recurrent?

Here https://math.stackexchange.com/questions/1019912/is-0-transient-positive-recurrent-or-null-recurrent

I already showed that $0$ is recurrent.

Now it remains to show whether $0$ is positive or null recurrent.

Therefore I have to show whether $$ \mathbb{E}_0(t(0))<\infty\text{ or }=\infty. $$

Here $t(0)$ denotes the first returning time of $0$.

How do I do this?

It is $$ \mathbb{E}_0(t(0))=\sum_{n\geq 1} 2n\mathbb{P}_0(t(0)=2n). $$

I am not sure how to compute $\mathbb{P}_0(t(0)=2n)$. I think it is $$ \mathbb{P}_0(t(0)=2n)=\frac{1}{4} (1-p_{00}^{2(n-1)}), $$ because it means starting in 0 one does not return to 0 within $2(n-1)$ steps and then going back to $0$ in 2 steps.

Is that right?

And if yes, how can I comoute the series then?

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mathfemi
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1 Answers1

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I am not sure how to compute $\mathbb{P}_0(t(0)=2n)$. I think it is $$ \mathbb{P}_0(t(0)=2n)=\frac{1}{4} (1-p_{00}^{2(n-1)}), $$

It is not.

After rereading the (exhausting) exchanges on the previous question and the comments here, it seems the question the OP has in mind is actually the following.

Assume we are given $P_0(X_n=0)$ for every $n\geqslant1$, for some irreducible recurrent Markov chain $(X_n)$ on a state space containing $0$, how to determine whether the chain is null recurrent or positive recurrent?

It happens that positive recurrent Markov chains visit the state $0$ roughly once in every $E_0(T_0)$ steps, in particular $\sum\limits_{k=1}^nP_0(X_k=0)$ grows linearly with respect to $n$.

In your case, $\sum\limits_nP_0(X_n=0)$ diverges and $P_0(X_n=0)\to0$ hence the chain is null recurrent.

Did
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  • I am not sure if I do understand that. As you know I want to know what $P_0(t(0)=2k)$ is. And I do not understand what you are doing in this answer to be honest. Why does this answer my problem? Sorry that I did not understand it yet. – mathfemi Nov 16 '14 at 11:05
  • "As you know I want to know what P0(t(0)=2k) is" No, for the nth time, the question is asking "whether 0 is positive or null recurrent" and the value of P0(t(0)=2k) for every k is not needed to do that. If you want to ask for P0(t(0)=2k) for every k, ask another question. – Did Nov 16 '14 at 12:56
  • Yes and my question is why I do not need to compute $P_0(t(0)=2k)$ in order to answer the question whether $0$ is positive or null recurrent; you wrote that in your answer, but I do not understand it; especially I do not understand why the questions reduces to that question that you wrote in your grey box. – mathfemi Nov 16 '14 at 13:01
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    I could fathom why somebody lacking all background on the elementary theory of Markov chains would need explanations for the assertions below the second grey box in my answer. But the second grey box itself happens to be no more than a trivial reformulation of your question, hence it is impossible that you do not understand it. Or rather, if really you do not understand the grey box then the whole thing is beyond my area of expertise--in any case, last comment from me. – Did Nov 16 '14 at 13:08
  • I do understand what is below the second grey boy but not why we need the assumption that $P_0(X_n=0)$ for all $n\geq 1$. – mathfemi Nov 16 '14 at 13:46