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.

中文翻译：

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复杂的采样设计：统一的极限定理和应用

在本文中，我们开发了一种通用方法来证明因复杂抽样设计而产生的Horvitz-Thompson实证过程的全局和局部一致极限定理。对于Horvitz-Thompson经验过程及其校准版本，都证明了诸如Glivenko-Cantelli和Donsker定理之类的全局定理，以及诸如局部渐近模量和相关比率类型极限定理之类的局部定理。还建立了其他变体的极限定理及其条件版本。我们的方法揭示了一个有趣的特征：一旦函数类满足通常的复杂性条件，那么为Horvitz-Thompson经验过程推导统一极限定理的问题就比建立相应的有限维极限定理的问题难得多。