  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.
