Moderate Deviation and Berry-Esseen Bounds in the $p$-Spin Curie-Weiss Model
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| Format: | Preprint |
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2024
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| _version_ | 1866910377381462016 |
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| author | Mukherjee, Somabha Liu, Tianyu Bhattacharya, Bhaswar B. |
| author_facet | Mukherjee, Somabha Liu, Tianyu Bhattacharya, Bhaswar B. |
| contents | Limit theorems for the magnetization in the $p$-spin Curie-Weiss model, for $p \geq 3$, has been derived recently by Mukherjee et al. (2021). In this paper, we strengthen these results by proving Cramér-type moderate deviation theorems and Berry-Esseen bounds for the magnetization (suitably centered and scaled). In particular, we show that the rate of convergence is $O(N^{-\frac{1}{2}})$ when the magnetization has asymptotically Gaussian fluctuations, and it is $O(N^{-\frac{1}{4}})$ when the fluctuations are non-Gaussian. As an application, we derive a Berry-Esseen bound for the maximum pseudolikelihood estimate of the inverse temperature in $p$-spin Curie-Weiss model with no external field, for all points in the parameter space where consistent estimation is possible. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_14122 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Moderate Deviation and Berry-Esseen Bounds in the $p$-Spin Curie-Weiss Model Mukherjee, Somabha Liu, Tianyu Bhattacharya, Bhaswar B. Probability Statistics Theory Limit theorems for the magnetization in the $p$-spin Curie-Weiss model, for $p \geq 3$, has been derived recently by Mukherjee et al. (2021). In this paper, we strengthen these results by proving Cramér-type moderate deviation theorems and Berry-Esseen bounds for the magnetization (suitably centered and scaled). In particular, we show that the rate of convergence is $O(N^{-\frac{1}{2}})$ when the magnetization has asymptotically Gaussian fluctuations, and it is $O(N^{-\frac{1}{4}})$ when the fluctuations are non-Gaussian. As an application, we derive a Berry-Esseen bound for the maximum pseudolikelihood estimate of the inverse temperature in $p$-spin Curie-Weiss model with no external field, for all points in the parameter space where consistent estimation is possible. |
| title | Moderate Deviation and Berry-Esseen Bounds in the $p$-Spin Curie-Weiss Model |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2403.14122 |