A martingale approach to Gaussian fluctuations and laws of iterated logarithm for Ewens-Pitman model
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| Format: | Preprint |
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2024
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| author | Bercu, Bernard Favaro, Stefano |
| author_facet | Bercu, Bernard Favaro, Stefano |
| contents | The Ewens-Pitman model refers to a distribution for random partitions of $[n]=\{1,\ldots,n\}$, which is indexed by a pair of parameters $α\in [0,1)$ 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 $K_{n}$ of partition sets and the number $K_{r,n}$ of partition subsets of size $r$, for $r=1,\ldots,n$. While for $α=0$ asymptotic results have been obtained in terms of almost-sure convergence and Gaussian fluctuations, for $α\in(0,1)$ only almost-sure convergences are available, with the proof for $K_{r,n}$ 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 $K_{n}$ and $K_{r,n}$ for $α\in(0,1)$, providing alternative, and rigorous, proofs of the almost-sure convergences of $K_{n}$ and $K_{r,n}$, and covering the gap of Gaussian fluctuations. We also obtain new laws of the iterated logarithm for $K_{n}$ and $K_{r,n}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_07694 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A martingale approach to Gaussian fluctuations and laws of iterated logarithm for Ewens-Pitman model Bercu, Bernard Favaro, Stefano Probability The Ewens-Pitman model refers to a distribution for random partitions of $[n]=\{1,\ldots,n\}$, which is indexed by a pair of parameters $α\in [0,1)$ 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 $K_{n}$ of partition sets and the number $K_{r,n}$ of partition subsets of size $r$, for $r=1,\ldots,n$. While for $α=0$ asymptotic results have been obtained in terms of almost-sure convergence and Gaussian fluctuations, for $α\in(0,1)$ only almost-sure convergences are available, with the proof for $K_{r,n}$ 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 $K_{n}$ and $K_{r,n}$ for $α\in(0,1)$, providing alternative, and rigorous, proofs of the almost-sure convergences of $K_{n}$ and $K_{r,n}$, and covering the gap of Gaussian fluctuations. We also obtain new laws of the iterated logarithm for $K_{n}$ and $K_{r,n}$. |
| title | A martingale approach to Gaussian fluctuations and laws of iterated logarithm for Ewens-Pitman model |
| topic | Probability |
| url | https://arxiv.org/abs/2404.07694 |