Novel asymptotical results for Ewens–Pitman partitions, with statistical applications

This thesis studies the large-sample behavior of the Ewens-Pitman model for random partitions under a non-standard asymptotic regime in which the concentration parameter θ grows linearly with the sample size. The Ewens–Pitman model is a two-parameter generalization of the classical Ewens model and is widely used in probability, population genetics, and Bayesian nonparametrics. While the asymptotic properties of the model are well understood when the parameters are fixed, much less is known when the concentration parameter increases with the sample size. The thesis focuses on the large-theta regime, where the concentration parameter is proportional to the sample size, i.e. θ=λn for some λ>0. In this setting, strong laws of large numbers and Gaussian central limit theorems are established for the number of blocks in the partition for all values of the parameter ɑ∈[0,1). These results show that, in contrast to the standard regime, the number of blocks grows linearly with the sample size and converges to a deterministic limit for all values of ɑ. Different proof techniques are developed depending on whether ɑ=0 or ɑ∈(0,1), including representations based on Bernoulli sums and compound Poisson constructions. An alternative, unified proof based on martingale methods is also introduced. A generalisation of such proof also allows to further extend the analysis to joint asymptotic results for the numbers of blocks of fixed sizes, providing a finer description of the partition structure. The thesis then applies these results to Bayesian nonparametric inference, deriving Gaussian asymptotic approximations for the posterior distribution of the number of unseen species and constructing efficient credible intervals. Finally, possible generalizations of the results obtained in the thesis are discussed, such as the possibility to consider further non-standard asymptotic regimes, or to extend the asymptotic analysis to other variables of interest beyond the numbers of blocks of the partition. Overall, the thesis advances the theoretical understanding of Ewens--Pitman random partitions under non-standard asymptotic scalings and illustrates how such probabilistic insights can be applied to statistical problems, expecially in the field of Bayesian nonparametrics.