A Rigorous Analysis of the Clauser-Horne-Shimony-Holt Inequality Experiment When Trials Need Not Be Independent

The Clauser-Horne-Shimony-Holt (CHSH) inequality is a constraint that local theories must obey. Quantum Mechanics predicts a violation of this inequality in certain experimental settings. Treatments of this subject frequently make simplifying assumptions about the probability spaces available to a local hidden variable theory, such as assuming the state of the system is a discrete or absolutely continuous random variable, or assuming that repeated experimental trials are independent and identically distributed. In this paper, we do two things: first, show that the CHSH inequality holds even for completely general state variables in the measure-theoretic setting, and second, demonstrate how to drop the assumption of independence of subsequent trials while still being able to perform a hypothesis test that will distinguish Quantum Mechanics from local theories. The statistical strength of such a test is computed.