For a joint model-based and design-based inference, we establish functional central limit theorems for the Horvitz–Thompson empirical process and the Hájek empirical process centered by their finite population mean as well as by their super-population mean in a survey sampling framework. The results apply to single-stage unequal probability sampling designs and essentially only require conditions on higher order correlations. We apply our main results to a Hadamard differentiable statistical functional and illustrate its limit behavior by means of a computer simulation.
"Functional central limit theorems for single-stage sampling designs." Ann. Statist. 45 (4) 1728 - 1758, August 2017. https://doi.org/10.1214/16-AOS1507