  Frequentists' inference often delivers point estimators associated with
confidence intervals or sets for parameters of interest. Constructing the
confidence intervals or sets requires understanding the sampling distributions
of the point estimators, which, in many but not all cases, are related to
asymptotic Normal distributions ensured by central limit theorems. Although
previous literature has established various forms of central limit theorems for
statistical inference in super population models, we still need general and
convenient forms of central limit theorems for some randomization-based causal
analysis of experimental data, where the parameters of interests are functions
of a finite population and randomness comes solely from the treatment
assignment. We use central limit theorems for sample surveys and rank
statistics to establish general forms of the finite population central limit
theorems that are particularly useful for proving asymptotic distributions of
randomization tests under the sharp null hypothesis of zero individual causal
effects, and for obtaining the asymptotic repeated sampling distributions of
the causal effect estimators. The new central limit theorems hold for general
experimental designs with multiple treatment levels and multiple treatment
factors, and are immediately applicable for studying the asymptotic properties
of many methods in causal inference, including instrumental variable,
regression adjustment, rerandomization, clustered randomized experiments, and
so on. Previously, the asymptotic properties of these problems are often based
on heuristic arguments, which in fact rely on general forms of finite
population central limit theorems that have not been established before. Our
new theorems fill in this gap by providing more solid theoretical foundation
for asymptotic randomization-based causal inference.
