I have a process with indices in $\mathbb{Z}$ with the following transition probabilities: $P(i,i+2)=p$ and $P(i,i-1)=1-p$, with $p\in(0,1)$, i.e. with probability $p$ I take two steps forward and with probability $1-p$ I go one step backwards. I have to determine for which values of $p$ this Markov chain is recurrent.
My idea was to compute the powers of the transition matrix (it´s easy to notice that the chain has period 3, so one is interested in the diagonal of the matrices $P^{3n}$), then check with the criterion saying that the chain is recurrent iff $\sum_{n}P^n(i,i)=\infty$. However, the computations seem cumbersome and I was wondering if anyone has a better idea on how to do this. A reference (on this process) would also be extremely appreciated.
To see why for $p \neq 1/3$ the process is transient, just apply Law of Large Numbers.
Let $X_i \in \{ -1, 2 \}$ where $X_i = -1 $(with probability $1-p$) and $X_i = 2$ (with probability $p$). Then, we are talking about the process $(S_n)_n$ where $S_n = \sum_{i=1}^n X_i $, right?
Well, we have $\mathbb{E}[X_i] = 3p-1$, thus, using independence of $X_i$'s, Law of Large Numbers implies that
$$ \frac{S_n}{n} \longrightarrow 3p-1 $$ almost surely. This means that your process has a drift to the right or the left (depending on $p$). This also implies that if you start the chain at zero, for example, you will visit it only a finite number of times and since the chain is irreducible, you have that the only chance of being recurrent is if $p=1/3$.
For $p=1/3$ you can apply similar argument employed to prove that the symmetric random walk is recurrent.
Hope this can help!