Sampling Mixtures of Multi-Sized Particles: An Application of Renewal Theory

It is assumed that a population consists of r(r < ∞) types of items, each item being characterized by a real-valued, non-negative random variable called the item's amount. Each type of item may have a different distribution of amounts. It is also assumed that samples are selected from this population so as to contain a fixed total amount rather than a fixed number of items. Finding the asymptotic distribution of the sample amounts for the different types of items, estimating the population amount proportions, and estimating the average item amount are three problems for which answers are found by use of methods and theorems from renewal theory. Some results which have already appeared in the literature are derived and extended in the course of this work.