I'm trying to find out what is known about time-inhomogeneous ergodic Markov Chains where the transition matrix can vary over time. All textbooks and lecture notes I could find initially introduce Markov chains this way but then quickly restrict themselves to the time-homogeneous case where you have one transition matrix.
Obviously, in general such Markov chains might not converge to a unique stationary distribution, but I would be surprised if there isn't a large (sub)class of these chains where convergence is guaranteed. I'm particularly interested in theorems on the mixing time and convergence theorems that state when there exists a stationary distribution.
Best Answer
You don't find much about time-inhomogeneous Markov chains because it's extremely difficult to prove anything about them without strong additional assumptions, and it's not clear what additional assumptions make sense. The only literature I'm aware of is some quite recent papers by Jessica Zúñiga and Laurent Saloff-Coste; see this page for links. (If there is any prior literature on the problems you're interested in, I'm sure those papers have references.)