Convergence of an inhomogeneous markov chain

Question

A markov chain is defined as $X_t=F(X_{t-1})X_{t-1}$, where $X_t$ and $X_{t-1}$ are both vector. So the transition matrix depends on the current states. I want to show that for any given initial states, the markov convergences to the same steady states with some known properties of $F(\cdot)$. I was trying to find some references about this issues and hardly find some useful ones. Can anyone provide some paper or whatever references addressing this kind of problem?

Answer 1

There might not be a steady state, in general. Mixing is the key notion that can still guarantee convergence in the non-homogeneous case. Here is one result out of many: http://arxiv.org/abs/0807.4665

See also this related question: Time-inhomogeneous Markov Chains