Design-Based Inference under Random Potential Outcomes

We develop a design-based framework for causal inference that accommodates random potential outcomes without introducing outcome models, thereby extending the classical Neyman–Rubin paradigm in which outcomes are treated as fixed. By modelling potential outcomes as random functions driven by a latent stochastic environment, causal estimands are defined as expectations over this mechanism rather than as functionals of a single realised potential-outcome schedule. We show that under local dependence, cross-sectional averaging exhibits an ergodic property that links a single realised experiment to the underlying stochastic mechanism, providing a fundamental justification for using classical design-based statistics to conduct inference on expectation-based causal estimands. We establish consistency, asymptotic normality, and feasible variance estimation for aggregate estimators under general dependency graphs. Our results clarify the conditions under which design-based inference extends beyond realised potential-outcome schedules and remains valid for mechanism-level causal targets.