Empirical Processes in Survey Sampling with (Conditional) Poisson Designs

It is the main purpose of this paper to study the asymptotics of certain variants of the empirical process in the context of survey data. Precisely, Functional Central Limit Theorems are established under usual conditions when the sample is drawn from a Poisson or a rejective sampling design. The framework we develop encompasses sampling designs with non-uniform first order inclusion probabilities, which can be chosen so as to optimize estimation accuracy. Applications to Hadamard differentiable functionals are considered.