I have a problem with a Markov chain that is time-inhomogeneous. Consider a Markov chain $(X_t)$ in discrete time $t=0, 1, \dotsc$ on the integers. At each time $t$, the process can go up by 1, down by 1 or stay constant, but these probabilities depend on time $t$. Formally, \begin{align*} \mathbb{P}[X_{t+1}=X_t+1|X_t]=p_t;\\ \mathbb{P}[X_{t+1}=X_t-1|X_t]=q_t;\\ \mathbb{P}[X_{t+1}=X_t|X_t]=1-p_t- q_t. \end{align*}
I am interested in the recurrence properties of the processes $(X_t)_t$ and how they relate to the probabilities $(p_t, q_t)_t$. Unfortunately, I did not find a good, accessible reference. Most textbooks focus on time-homogeneous Markov chains only.
Does anybody have a suggestion for a good reference?