Comprehensive reference for lumped or projected markov chains

Consider a Markov chain $X_n$ taking values in finite countable set $\mathcal{X}$ with transition matrix $P$. Consider a function $f:\mathcal{X}\to\mathcal{Y}$ inducing a partition $\mathcal{Y}=\{\mathcal{X}_{[i]}\}$. Then, the process $Y_n = f(X_n)$ is Markov if, given any two elements of the partition $\mathcal{X_{[k]}}$ and $\mathcal{X_{[\ell]}}$ it holds

$$\sum_{z\in\mathcal{X_{[\ell]}}}P(x\to z)=\sum_{z\in\mathcal{X}_{[\ell]}}P(x'\to z),\ \text{for all}\ x,x'\in \mathcal{X}_{[k]}.$$

I would like to know if there exists a comprehensive reference on the properties preserved under this type of reduction/coarsening of the state space, which is usually called lumping or projection.

I am particularly interested to this in relationship to particle systems with finite state space and coupling techniques for such systems, but if there's anything more general or on different type of spaces that would be welcome as well.

• These are commonly called hidden Markov models. I'm not an expert on such models but there should be a vast literature. Feb 16 '18 at 22:16
• I believe that the hidden Markov model terminology is typically used for applications (simulation, inference and parameter estimates), whereas within probability theory they are referred to as lumped or projected chains instead (and the questions are of course different: is the model still Markov would be the prototypical one). Feb 19 '18 at 13:09