TITLE: Time-inhomogeneous Markov chains
QUESTION [16 upvotes]: 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.

REPLY [2 votes]: There is also the (old) book of Seneta: Non-negative Matrices and Markov Chains.
First edition: 1973. See chapters 3-4.