Laws of large numbers and central limit theorem for Ewens-Pitman model

The Ewens-Pitman model is a distribution on random partitions of {1, …, n}, with n ∈ N, indexed by α ∈ [0, 1) and θ > −α, where α = 0 corresponds to the Ewens model in population genetics. The large n asymptotic behaviour of the number K_n of blocks in the Ewens-Pitman random partition has been extensively studied, showing that K_n scales as log n for α=0 and as n^α for α ∈ (0,1), yielding a non-random limit and random limit, respectively. In this paper, we study the large n asymptotic behaviour of K_n when the parameter θ is allowed to depend linearly on n ∈ N, namely θ = λn with λ > 0. This non-standard asymptotic regime first appeared in Feng (The Annals of Applied Probability, 17, 2007) for the special case α=0, for which a law of large numbers (LLN) and a central limit theorem (CLT) were later established. We extend these results to the general case α ∈ (0, 1), showing that K_n scales as n for all α ∈ [0, 1), yielding non-random limits. The CLTs rely on different techniques depending on whether α = 0 or α ∈ (0, 1). For α = 0, we provide an alternative proof of the CLT based on representing K_n as a sum of independent, but not identically distributed, Bernoulli random variables, which also yields a Berry-Esseen theorem for K_n. Instead, for α ∈ (0, 1), we rely on a compound Poisson construction of K_n, leading to prove a LLN, a CLTs and a Berry-Esseen theorem for the number of blocks of the negative-Binomial compound Poisson random partition, results of independent interest.