Local Dependence Graphs for Discrete Time Processes

Local dependence graphs for discrete time processes encapsulate information concerning the dependence relationships between the past of the multidimensional process and its present state and as such can represent feedback loops. Even in the discrete time setting some natural questions relating the conditional (in)dependence statements in the stochastic process to separation properties of the underlying local dependence graph are scattered throughout the literature. We provide an unifying view and fill in certain gaps. In this paper we examine graphical characteristics for two kinds of conditional independences: those occurring in Markov chains under the stationary regime and independences between the past of one subprocess and the future of another given the past of the third subprocess.