Reciprocal Chains: Foundations

Reciprocal processes are stochastic processes such that the current state only depends on the nearest past and future. The research on continuous-time reciprocal processes has been comprehensive since the 1970s, while surprisingly discrete-time reciprocal processes (i.e., reciprocal chains) are rarely seen in the literature. What is worse is that in the scant literature where reciprocal chains are mentioned, a misunderstanding on how to formally define reciprocal chains has existed since 2008. This article aims to formally define reciprocal chains and provide foundations for the study of them.