arXiv subjects

Claudia Contardi

Publications and source records attributed to Claudia Contardi.

3 recordsLinked to original sources

arXiv2026

A Central Limit Theorem for the Ewens-Pitman random partition in the large-$\theta$ regime via a martingale approach

The Ewens-Pitman model defines a distribution on random partitions of $\{1,\ldots,n\}$, with parameters $\alpha \in [0,1)$ and $\theta > -\alpha$; the case $\alpha=0$ reduces to the classical Ewens model from population genetics. We investigate the large-$n$ asymptotic behaviour of the Ewens-Pitman random partition in the nonstandard regime $\theta=\lambda n$ with $\lambda>0$, establishing joint fluctuation results for the total number of blocks $K_n^{\{n\}}$ and the counts $K_{r,n}^{\{n\}}$ of blocks of sizes $r=1,\dots,d$, for fixed $d\in\mathbb{N}$. In particular, for $\alpha\in[0,1)$ and $\theta=\lambda n$, our main result provides a strong law of large numbers and a central limit theorem for the $(d+1)$-dimensional vector $\mathbf{K}_{d,n}^{\{n\}} = \bigl(K_n^{\{n\}}, K_{1,n}^{\{n\}}, \dots, K_{d,n}^{\{n\}}\bigr)^T$ as $n \to \infty$. The proof exploits the Chinese restaurant sequential construction under $\theta=\lambda n$ and a central limit theorem for triangular arrays of martingales, extending techniques previously developed for the classical regime with fixed $\theta$. As corollaries of our results, we recover known asymptotics for $K_n^{\{n\}}$ and derive new strong laws and central limit theorems for each fixed $K_{r,n}^{\{n\}}$, thereby completing earlier weak-law results and providing a comprehensive asymptotic description of the Ewens-Pitman partition structure in the large-$\theta$ setting.

math.PR↗

arXiv2025

Gaussian credible intervals in Bayesian nonparametric estimation of the unseen

The unseen-species problem assumes $n\geq1$ samples from a population of individuals belonging to different species, possibly infinite, and calls for estimating the number $K_{n,m}$ of hitherto unseen species that would be observed if $m\geq1$ new samples were collected from the same population. This is a long-standing problem in statistics, which has gained renewed relevance in biological and physical sciences, particularly in settings with large values of $n$ and $m$. In this paper, we adopt a Bayesian nonparametric approach to the unseen-species problem under the Pitman-Yor prior, and propose a novel methodology to derive large $m$ asymptotic credible intervals for $K_{n,m}$, for any $n\geq1$. By leveraging a Gaussian central limit theorem for the posterior distribution of $K_{n,m}$, our method improves upon competitors in two key aspects: firstly, it enables the full parameterization of the Pitman-Yor prior, including the Dirichlet prior; secondly, it avoids the need of Monte Carlo sampling, enhancing computational efficiency. We validate the proposed method on synthetic and real data, demonstrating that it improves the empirical performance of competitors by significantly narrowing the gap between asymptotic and exact credible intervals for any $m\geq1$.

stat.ME↗

arXiv2024

Laws of large numbers and central limit theorem for Ewens-Pitman model

The Ewens-Pitman model is a distribution for random partitions of the set $\{1,\ldots,n\}$, with $n\in\mathbb{N}$, indexed by parameters $\alpha \in [0,1)$ and $\theta>-\alpha$, such that $\alpha=0$ is the Ewens model in population genetics. The large $n$ asymptotic behaviour of the number $K_{n}$ of blocks in the Ewens-Pitman random partition has been extensively investigated in terms of almost-sure and Gaussian fluctuations, which show that $K_{n}$ scales as $\log n$ and $n^{\alpha}$ depending on whether $\alpha=0$ or $\alpha\in(0,1)$, providing non-random and random limiting behaviours, respectively. In this paper, we study the large $n$ asymptotic behaviour of $K_{n}$ when the parameter $\theta$ is allowed to depend linearly on $n\in\mathbb{N}$, a non-standard asymptotic regime first considered for $\alpha=0$ in Feng (\textit{The Annals of Applied Probability}, \textbf{17}, 2007). In particular, for $\alpha\in[0,1)$ and $\theta=\lambda n$, with $\lambda>0$, we establish a law of large numbers (LLN) and a central limit theorem (CLT) for $K_{n}$, which show that $K_{n}$ scales as $n$, providing non-random limiting behaviours. Depending on whether $\alpha=0$ or $\alpha\in(0,1)$, our results rely on different arguments. For $\alpha=0$ we rely on the representation of $K_{n}$ 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 $\alpha\in(0,1)$, we rely on a compound Poisson construction of $K_{n}$, 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.

math.PR↗