Let $\{X_t\}_{t=-\infty}^{\infty}$ be a sequence of random variables. We are interested in a "random walk" (or more generally, a random field) that can be characterized by $$ X_t ~|~ X_{t-k}, \ldots, X_{t-1}, X_{t+1}, \ldots, X_{t+k} \sim D_t, $$ where $k>0$ is a constant and $D_t$ is some distribution. Here we assume the joint distribution of $\{X_t\}_{t=-\infty}^{\infty}$ exists. This "random walk" is quite similar to the classical Markovian random walk, but here we have "forward dependency" on $X_{t+1}, \ldots, X_{t+k}$.

I am wondering if this kind of "random walk" is studied in the literature. Is there a possible way to transform it into a Markovian random walk? In particular, can we use the notion of weak dependency and mixing time to characterize the concentration of the functionals of $X_t$?