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 $X\_t$?



Random Walk with "Forward Dependency"
if we assume that the index $t$ in your process is countable then what you are looking for is described in Georgii's book, Gibbs Measures and Phase Transition. In the language of mathematical Statistical Mechanics the processes you are interested in is a one-dimensional spin system on the lattice $\mathbb{Z}$ with finite range interactions. Georgii discuss extistence and uniqueness of such processes even in the case $k$ is infinity. For one-sided sequences this is also studied in Ergodic Theory and called Thermodynamic Formalism. In this setting normally $X\_t$ is supposed to take values in a compact set. A good reference for one-sided sequences is the book of Parry and Pollicott. 


