Asymptotic Theory for Successive Sampling with Varying Probabilities Without Replacement, I

To each of the items 1,2,\⋯, N in a finite population there is associated a variate value. The population is sampled by successive drawings without replacement in the following way. At each draw the probability of drawing item s is proportional to a number pₛ > 0 if item s remains in the population and is 0 otherwise. Let \Δ(s; n) be the probability that item s is obtained in the first n draws and let Zₙ be the sum of the variate values obtained in the first n draws. Asymptotic formulas, valid under general conditions when n and N both are "large", are derived for \Δ(s; n), EZₙ and (Zn₁, Zn₂). Furthermore it is shown that, still under general conditions, the joint distribution of Zn₁, Zn₂,\⋯, Znd is asymptotically normal. The general results are then applied to obtain asymptotic results for a "quasi"-Horvitz-Thompson estimator of the population total.