  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. These global
and local uniform limit theorems are then applied to important statistical
problems including (i) $M$-estimation (ii) $Z$-estimation (iii) frequentist
theory of Bayes procedures, all with weighted likelihood, to illustrate their
wide applicability.
