# Functional central limit theorems for conditional Poisson sampling

## TL;DR

This paper refines and generalizes functional central limit theorems for conditional Poisson sampling, providing detailed proofs and insights useful for applications in survey sampling.

## Contribution

It offers more suitable, generalized versions of existing theorems with detailed proofs, enhancing understanding of weak convergence in survey sampling.

## Key findings

- Refined functional central limit theorems for conditional Poisson sampling.
- Detailed discussion on proving weak convergence in bounded function spaces.
- Enhanced theoretical framework for applications in survey sampling.

## Abstract

This paper provides refined versions of some known functional central limit theorems for conditional Poisson sampling which are more suitable for applications. The theorems presented in this paper are generalizations of some results that have been recently published by \citet*{Bertail_2017}. The asymptotic equicontinuity part of the proofs presented in this paper is based on the same idea as in \citep{Bertail_2017} but some of the missing details are provided. On the way to the functional central limit theorems, this paper provides a detailed discussion of what must be done in order to prove conditional and unconditional weak convergence in bounded function spaces in the context of survey sampling. The results from this discussion can be useful to prove further weak convergence results.