Suppose that we have an ergodic finite Markov chain $C$ with a fintie state space $S$, and we have random variables $X_s$ where $s\in S$. Consider the following random walk
$S_0=0$ and $S_{i+1}=S_i +X_{C_i}$.
(In words, the random walk $S$ is somehow time-inhomogeneous.)
Is the property of this random walk well studied? Can I get some reference. Thanks.