I am looking for references on Markov chains of the following type: the state space is $\mathbb{N}$, and the transition matrix is such that
- $0$ is absorbing;
- for any $x \in \mathbb{N}$, only $p_{x,x+1}$ and $p_{x,x-1}$ of the $p_{x,y}$'s are nonzero
- there exists $k \in \mathbb{N}^*$ such that for all $x \in \mathbb{N}$, $p_{x,x+1} = p_{x+k,x+k+1}$.
These Markov chains are therefore defined by the finite data ($(p_{x,x+1})_{x \in \{1,\cdots,k\}}$ for a relevant $k$).
I would also like to have a description of the asymptotic behaviour of these Markov chains in terms of $(p_{x,x+1})_{x \in \{1,\cdots,k\}}$.