A Martingale Approach to Large-$θ$ Ewens-Pitman Model
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915213245153280 |
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| author | Ribeiro, Rodrigo |
| author_facet | Ribeiro, Rodrigo |
| contents | We investigate the asymptotic behavior of the number of parts $K_n$ in the Ewens--Pitman partition model under the regime where the diversity parameter is scaled linearly with the sample size, that is, $θ= λn$ for some~$λ> 0$. While recent work has established a law of large numbers (LLN) and a central limit theorem (CLT) for $K_n$ in this regime, we revisit these results through a martingale-based approach. Our method yields significantly shorter proofs, and leads to sharper convergence rates in the CLT, including improved Berry--Esseen bounds in the case $α= 0$, and a new result for the regime $α\in (0,1)$, filling a gap in the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_19892 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Martingale Approach to Large-$θ$ Ewens-Pitman Model Ribeiro, Rodrigo Probability Statistics Theory We investigate the asymptotic behavior of the number of parts $K_n$ in the Ewens--Pitman partition model under the regime where the diversity parameter is scaled linearly with the sample size, that is, $θ= λn$ for some~$λ> 0$. While recent work has established a law of large numbers (LLN) and a central limit theorem (CLT) for $K_n$ in this regime, we revisit these results through a martingale-based approach. Our method yields significantly shorter proofs, and leads to sharper convergence rates in the CLT, including improved Berry--Esseen bounds in the case $α= 0$, and a new result for the regime $α\in (0,1)$, filling a gap in the literature. |
| title | A Martingale Approach to Large-$θ$ Ewens-Pitman Model |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2503.19892 |