  The Ewens-Pitman model refers to a distribution for exchangeable 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
celebrated 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 sets that appear $r$ times, 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 $K_{r,m}$ 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 $K_{r,n}$.
