# Random Walk with "Forward Dependency"

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 $X_t$?

• One immediate (maybe naive) reason these objects are undesirable to study is that your class of stopping times becomes greatly restricted: if your random walk is non-causal then you can no longer decide when to stop based on the walk's history. What is gained by interpreting its index as temporal? Maybe what you have in mind is an object like a Brownian sheet, or some process over a non-linearly-ordered domain. But I would not call such a thing a 'random walk'. Commented Sep 6, 2017 at 9:38
• @enthdegree Yes good point. I have revised the question accordingly. Commented Sep 7, 2017 at 0:32

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.