I want to calculate for which values of $p$ and $q$ the Markov chain above is recurrent , positive recurrent and transient. To do this, I primarily tried reducing the Markov chain to a simpler one by bundling states together, with the aim of reducing the problem to a gambler's ruin problem. The only problem is that when I bundle states together, e.g. {1,1'}, {2,2'}, etc, the transition probabilities between layers are not the same for each state in the bundle.
I also tried using the criterion that a state i is recurrent iff $\sum_{n=0}^{n=\infty}{p_{nn}(i)} = \infty$. However this got messy quite fast, so I think I am missing something obvious for the first idea.
Any hints would be greatly appreciated. If I still cannot solve it with a hint I might ask for extra help.
Best Answer
Use two appropriate stopping times, and find an equation linking the probability they are less than infinity.