How to show positive recurrence/ null recurrence?

Question

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?

Answer 1

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.