Suppose a Markov chain with $p_{ii+2}=\frac{3}{5}$ and $p_{ii-3}=\frac{2}{5}$, $i\in \mathbb{Z}$, then how to prove that this markov chain is recurrent and irreducible?
I try to prove that $\sum p_{ii}^n=\infty$, but I can't calculate the $p_{ii}^n$.
You return to $i$ after $n$ steps if $n$ is a multiple of $5$ and $2n/5$ steps were to the left and $3n/5$ steps were to the right. When $n=5k$, this occurs with probability ${5k \choose 2k} \left ( \frac{2}{5} \right )^{2k} \left ( \frac{3}{5} \right )^{3k}$. Now use Stirling's approximation to estimate the binomial coefficient, and you should find a non-summable lower bound for $p_{ii}^n$.