2024/12/16 by Contardi, Claudia, Dolera, Emanuele, Favaro, Stefano · 1 citation
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2412.11493
The Ewens-Pitman model is a distribution for random partitions of the set \1,…,n\, with n∈ℕ, indexed by parameters α∈ [0,1) and θ>-α, such that α=0 is the Ewens model in population genetics. The large n asymptotic behaviour of the number Kn of blocks in the Ewens-Pitman random partition has been extensively investigated in terms of almost-sure and Gaussian fluctuations, which show that Kn scales as log n and nα depending on whether α=0 or α∈(0,1), providing non-random and random limiting behaviours, respectively. In this paper, we study the large n asymptotic behaviour of Kn when the parameter θ is allowed to depend linearly on n∈ℕ, a non-standard asymptotic regime first considered for α=0 in Feng (The Annals of Applied Probability, 17, 2007). In particular, for α∈[0,1) and θ=λn, with λ>0, we establish a law of large numbers (LLN) and a central limit theorem (CLT) for Kn, which show that Kn scales as n, providing non-random limiting behaviours. Depending on whether α=0 or α∈(0,1), our results rely on different arguments. For α=0 we rely on the representation of Kn as a sum of independent, but not identically distributed, Bernoulli random variables, which leads to a refinement of the CLT in terms of a Berry-Esseen theorem. Instead, for α∈(0,1), we rely on a compound Poisson construction of Kn, leading to prove LLNs, CLTs and Berry-Esseen theorems for the number of blocks of the negative-Binomial compound Poisson random partition, which are of independent interest.