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 $[n]=\{1,\ldots,n\}$, 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 \textcolor{blue}{the number $K_{r,n}$ of partition subsets of size $r$}, for $r=1,\ldots,n$. 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 $K_{r,n}$.