Moderate Deviations for Ewens-Pitman Sampling Models

Consider a population of individuals belonging to an infinity number of types, and assume that type proportions follow the Poisson-Dirichlet distribution with parameter α ∈ [0,1) and 𝜃 > −α. Given a sample of size n from the population, two important statistics are the number Kn of different types in the sample, and the number Ml,n of different types with frequency l in the sample. We establish moderate deviation principles for (Kn)n≥ 1 and (Ml,n)n≥ 1. Corresponding rate functions are explicitly identified, which help in revealing a critical scale and in understanding the exact role of the parameters α and 𝜃.