This paper is a direct continuation of the corresponding paper , in which our main results were formulated in Section 3. Some of these results were proved and we prepared ourselves for the proof of the asymptotic normality of the sample sum. Our main concern in this part is to carry through the remaining proofs.
"Asymptotic Theory for Successive Sampling with Varying Probabilities Without Replacement, II." Ann. Math. Statist. 43 (3) 748 - 776, June, 1972. https://doi.org/10.1214/aoms/1177692543