A martingale approach to Gaussian fluctuations and laws of iterated logarithm for Ewens–Pitman model

The Ewens–Pitman model refers to a distribution for random partitions of $\left[n\right]=\{1,\dots ,n\}$, which is indexed by a pair of parameters $\alpha \in [0,1)$ and $\theta >-\alpha$, with $\alpha =0$ corresponding to the Ewens model in population genetics. The large $n$ asymptotic properties of the Ewens–Pitman model have been the subject of numerous studies, with the focus being on the number $K_n$ of partition sets and the number $K_{r,n}$ of partition subsets of size $r$, for $r=1,\dots ,n$. While for $\alpha =0$ asymptotic results have been obtained in terms of almost-sure convergence and Gaussian fluctuations, for $\alpha \in (0,1)$ only almost-sure convergences are available, with the proof for $K_{r,n}$ being given only as a sketch. In this paper, we make use of martingales to develop a unified and comprehensive treatment of the large $n$ asymptotic behaviours of $K_n$ and $K_{r,n}$ for $\alpha \in (0,1)$, providing alternative, and rigorous, proofs of the almost-sure convergences of $K_n$ and $K_{r,n}$, and covering the gap of Gaussian fluctuations. We also obtain new laws of the iterated logarithm for $K_n$ and $K_{r,n}$.