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Frequentists' inference often delivers point estimators associated withconfidence intervals or sets for parameters of interest. Constructing theconfidence intervals or sets requires understanding the sampling distributionsof the point estimators, which, in many but not all cases, are related toasymptotic Normal distributions ensured by central limit theorems. Althoughprevious literature has established various forms of central limit theorems forstatistical inference in super population models, we still need general andconvenient forms of central limit theorems for some randomization-based causalanalysis of experimental data, where the parameters of interests are functionsof a finite population and randomness comes solely from the treatmentassignment. We use central limit theorems for sample surveys and rankstatistics to establish general forms of the finite population central limittheorems that are particularly useful for proving asymptotic distributions ofrandomization tests under the sharp null hypothesis of zero individual causaleffects, and for obtaining the asymptotic repeated sampling distributions ofthe causal effect estimators. The new central limit theorems hold for generalexperimental designs with multiple treatment levels and multiple treatmentfactors, and are immediately applicable for studying the asymptotic propertiesof many methods in causal inference, including instrumental variable,regression adjustment, rerandomization, clustered randomized experiments, andso on. Previously, the asymptotic properties of these problems are often basedon heuristic arguments, which in fact rely on general forms of finitepopulation central limit theorems that have not been established before. Ournew theorems fill in this gap by providing more solid theoretical foundationfor asymptotic randomization-based causal inference.

General forms of finite population central limit theorems with applications to causal inference