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

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