Consistent ordered sampling distributions: characterization and convergence

<jats:p>This paper is concerned with models for sampling from populations in which there exists a total order on the collection of types, but only the relative ordering of types which actually appear in the sample is known. The need for consistency between different sample sizes limits the possible models to what are here called ‘consistent ordered sampling distributions'. We give conditions under which weak convergence of population distributions implies convergence of sampling distributions and conversely those under which population convergence may be inferred from convergence of sampling distributions. A central result exhibits a collection of ‘ordered sampling functions', none of which is continuous, which separates measures in a certain class. More generally, we characterize all consistent ordered sampling distributions, proving an analogue of de Finetti's theorem in this context. These results are applied to an unsolved problem in genetics where it is shown that equilibrium age-ordered population allele frequencies for a wide class of exchangeable reproductive models converge weakly, as the population size becomes large, to the so-called GEM distribution. This provides an alternative characterization which is more informative and often more convenient than Kingman's (1977) characterization in terms of the Poisson–Dirichlet distribution.</jats:p>