# Convergence of an inhomogeneous markov chain

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?