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

The Ewens-Pitman model refers to a distribution for exchangeable random partitions of [n]=\1,…,n\, which is indexed by a pair of parameters α∈[0,1) and θ>-α, with α=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 Kₙ of partition sets and the number Kr,n of partition sets that appear r times, for r=1,…,n. While for α=0 asymptotic results have been obtained in terms of almost-sure convergence and Gaussian fluctuations, for α∈(0,1) only almost-sure convergences are available, with the proof for Kr,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 Kₙ and Kr,n for α∈(0,1), providing alternative, and rigorous, proofs of the almost-sure convergences of Kₙ and Kr,n, and covering the gap of Gaussian fluctuations. We also obtain new laws of the iterated logarithm for Kₙ and Kr,n.