This is an exercise from Probability and Measure by Billingsley:
Let $(X_n)$ be a Markov chain with $\Bbb{Z}$ as its state space and with transition probabilities $(p_{ij})$ such that $$ \begin{align} p_{0,-1}&=p_{0,0}=p_{0,+1}=1/3,\\ p_{k,k-1}&=q,\ \ p_{k,k+1}=p,\quad k\leq -1,\\ p_{k,k-1}&=p,\ \ p_{k,k+1}=q,\quad k\geq 1, \end{align} $$ where $p+q=1$. Show that the chain is irreducible and aperiodic. For which $p$'s is the chain persistent? For which $p$'s are there stationary probabilities?
It's not difficult to show by definition that the chain is irreducible and aperiodic. The definition of "persistent" does not seem very helpful. What theorems would be needed and how should I go on?
One can note that the process $Y=|X|$ is also a Markov chain, but on the state space $\{0,1,2,\ldots\}$, with transitions $0\to0$ with probability $\frac13$, $0\to1$ with probability $\frac23$, $k\to k-1$ with probability $p$ and $k\to k+1$ with probability $q=1-p$, for every $k\geqslant1$.
You might be able to compute the type of $Y$ depending on $p$, then to deduce the type of $X$.