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

Abstract

The Ewens-Pitman model is a distribution for random partitions of the set $\{1,\ldots,n\}$, with $n\in\mathbb{N}$, indexed by parameters $α\in [0,1)$ and $θ>-α$, such that $α=0$ is 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 investigated in terms of almost-sure and Gaussian fluctuations, which show that $K_{n}$ scales as $\log n$ and $n^α$ depending on whether $α=0$ or $α\in(0,1)$, providing non-random and random limiting behaviours, respectively. In this paper, we study the large $n$ asymptotic behaviour of $K_{n}$ when the parameter $θ$ is allowed to depend linearly on $n\in\mathbb{N}$, a non-standard asymptotic regime first considered for $α=0$ in Feng (\textit{The Annals of Applied Probability}, \textbf{17}, 2007). In particular, for $α\in[0,1)$ and $θ=λn$, with $λ>0$, we establish a law of large numbers (LLN) and a central limit theorem (CLT) for $K_{n}$, which show that $K_{n}$ scales as $n$, providing non-random limiting behaviours. Depending on whether $α=0$ or $α\in(0,1)$, our results rely on different arguments. For $α=0$ we rely on the representation of $K_{n}$ as a sum of independent, but not identically distributed, Bernoulli random variables, which leads to a refinement of the CLT in terms of a Berry-Esseen theorem. Instead, for $α\in(0,1)$, we rely on a compound Poisson construction of $K_{n}$, leading to prove LLNs, CLTs and Berry-Esseen theorems for the number of blocks of the negative-Binomial compound Poisson random partition, which are of independent interest.