In this paper, we develop a general approach to proving global and local uniform limit theorems for the Horvitz–Thompson empirical process arising from complex sampling designs. Global theorems such as Glivenko–Cantelli and Donsker theorems, and local theorems such as local asymptotic modulus and related ratio-type limit theorems are proved for both the Horvitz–Thompson empirical process, and its calibrated version. Limit theorems of other variants and their conditional versions are also established. Our approach reveals an interesting feature: the problem of deriving uniform limit theorems for the Horvitz–Thompson empirical process is essentially no harder than the problem of establishing the corresponding finite-dimensional limit theorems, once the usual complexity conditions on the function class are satisfied. These global and local uniform limit theorems are then applied to important statistical problems including (i) $M$-estimation, (ii) $Z$-estimation and (iii) frequentist theory of pseudo-Bayes procedures, all with weighted likelihood, to illustrate their wide applicability.
"Complex sampling designs: Uniform limit theorems and applications." Ann. Statist. 49 (1) 459 - 485, February 2021. https://doi.org/10.1214/20-AOS1964