Cross-Sectional Conditional Independence in Stationary Time Series: Graphical Equivalence and Completeness of Collider Separation

We study cross-sectional conditional independence (CI) in strictly stationary time series represented by summary (local-dependence) graphs. Our focus is on the completeness of separation rules, i.e., when—and in what sense—graphical separation on the summary graph is not only sufficient, but also necessary for CI at a time slice. Prior work has established soundness of collider-based c-separation in discrete time and trek-graph separation in continuous time with respect to cross-sectional CI. We complement these results twofold. First, we show that, on summary graphs, c-separation and trek-graph separation induce the same equivalence classes of graphs. This identifies a single underlying graphical notion governing cross-sectional CI and reconciles discrete- and continuous-time separation rules. In addition, we translate c-separation to space-time graphs and introduce space-time trek separation completing the graphical separation picture. In essence, it is the collider structure that governs what shared past information travels to cross section through common ancestors and governs CI. Second, we prove the completeness of the collider separation for the discrete-time case. For any given summary graph, there exists a strictly stationary process whose cross-sectional law realizes all and only the c-separation statements. Moreover, within the family of stationary Gaussian VAR(1) models, in the absence of c-separation, parameter values that nonetheless yield independence form a Lebesgue measure-zero set. Together, these results sharpen the theoretical underpinnings of graphical reasoning about cross-sectional CI in stationary time series and clarify when faithfulness-type assumptions for causal discovery are generically justified for snapshots of dynamic Bayesian networks.