A Central Limit Theorem for the Ewens-Pitman random partition in the large-$θ$ regime via a martingale approach
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| Autores principales: | , , , |
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| Formato: | Preprint |
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2026
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| _version_ | 1866912851534282752 |
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| author | Bercu, Bernard Contardi, Claudia Dolera, Emanuele Favaro, Stefano |
| author_facet | Bercu, Bernard Contardi, Claudia Dolera, Emanuele Favaro, Stefano |
| contents | The Ewens-Pitman model defines a distribution on random partitions of $\{1,\ldots,n\}$, with parameters $α\in [0,1)$ and $θ> -α$; the case $α=0$ 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 $θ=λn$ with $λ>0$, 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 $α\in[0,1)$ and $θ=λn$, 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 $θ=λn$ and a central limit theorem for triangular arrays of martingales, extending techniques previously developed for the classical regime with fixed $θ$. 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-$θ$ setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18935 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Central Limit Theorem for the Ewens-Pitman random partition in the large-$θ$ regime via a martingale approach Bercu, Bernard Contardi, Claudia Dolera, Emanuele Favaro, Stefano Probability The Ewens-Pitman model defines a distribution on random partitions of $\{1,\ldots,n\}$, with parameters $α\in [0,1)$ and $θ> -α$; the case $α=0$ 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 $θ=λn$ with $λ>0$, 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 $α\in[0,1)$ and $θ=λn$, 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 $θ=λn$ and a central limit theorem for triangular arrays of martingales, extending techniques previously developed for the classical regime with fixed $θ$. 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-$θ$ setting. |
| title | A Central Limit Theorem for the Ewens-Pitman random partition in the large-$θ$ regime via a martingale approach |
| topic | Probability |
| url | https://arxiv.org/abs/2601.18935 |