This question is to some degree a follow-up of this question.
Suppose we have a random-walk like Markov chain, i.e. state space is the set of all integers from $-\infty$ to $\infty$, the transition probability $P_{ij}$ is nonzero only when $j=i+1$ or $j=i-1$.
The difference from the random walk is that the transition probability is periodic in state space and periodic in time (for simplicity, let's consider a period 2 case, and hence it is inhomogeneous Markov chain) i.e.
At time $2t$, $P_{i,i+1}=p_1,P_{i+1,i+2}=p_2,\dots,P_{i+L-1,i+L}=p_L,P_{i+L,i+L+1}=p_1$, and none of $p_i$ is 0 or 1. At time $2t+1$, $P_{i,i+1}=q_1,P_{i+1,i+2}=q_2,\dots,P_{i+L-1,i+L}=q_L,P_{i+L,i+L+1}=q_1$, and none of $q_i$ is 0 or 1.
My question is: under what conditions (in terms of $\{p_i\}$) is this Markov chain recurrent or transient? (Is this question even meaningful?)
Is it possible to generalize the results to inhomogeneous Markov chain with general finite time period $M$?
Consider $Y_n=X_{2n}$. Then $(Y_n)$ is a time-homogenous Markov chain. Its transition probabilities $R_{ij}$ are nonzero only for $j$ in $\{i-2,i,i+2\}$ and they are periodic in the sense that, for every $(i,j)$, $R_{i+L,j+L}=R_{i,j}$. The usual tools apply to $(Y_n)$. Finally, $(X_n)$ is positive recurrent / recurrent null / transient if and only if $(Y_n)$ is.