Is this random walk well studied?

Question

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.

Answer 1

This model is called random walk in random scenery. It was introduced by Harry Kesten and Frank Spitzer in 1979 in their paper A limit theorem related to a new class of self similar processes and is an active research subject: see the book Random Walk in Random And Non-Random Environments by Pál Révész, and the survey by Frank den Hollander and Jeffrey E. Steif, somewhat more oriented towards the ergodic properties.

Answer 2

Look for markov random walk or markov renewal process. It depends what you are interested in, but that latter are discussed briefly in Ross' probability models.