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

Bernard Bercu and Stefano Favaro

Stochastic Processes and their Applications, 2024, vol. 178, issue C

Abstract: The Ewens–Pitman model refers to a distribution for 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 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 subsets of size r, 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,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 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.

Keywords: Almost-sure limit; Ewens–Pitman model; Exchangeable random partition; Gaussian fluctuation; Law of iterated logarithm; Martingale; Mittag-Leffler distribution (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1016/j.spa.2024.104493

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