Rényi Divergences in Central Limit Theorems: Old and New
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| Format: | Preprint |
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2025
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| _version_ | 1866909526962208768 |
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| author | Bobkov, S. G. Chistyakov, G. P. Götze, F. |
| author_facet | Bobkov, S. G. Chistyakov, G. P. Götze, F. |
| contents | We give an overview of various results and methods related to information-theoretic distances of Rényi type in the light of their applications to the central limit theorem (CLT). The first part (Sections 1-9) is devoted to the total variation and the Kullback-Leibler distance (relative entropy). In the second part (Sections 10-15) we discuss general properties of Rényi and Tsallis divergences of order $α>1$, and then in the third part (Sections 16-21) we turn to the CLT and non-uniform local limit theorems with respect to these strong distances. In the fourth part (Sections 22-31), we discuss recent results on strictly subgaussian distributions and describe necessary and sufficient conditions which ensure the validity of the CLT with respect to the Rényi divergence of infinite order. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_03926 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rényi Divergences in Central Limit Theorems: Old and New Bobkov, S. G. Chistyakov, G. P. Götze, F. Information Theory Probability 60E, 60F We give an overview of various results and methods related to information-theoretic distances of Rényi type in the light of their applications to the central limit theorem (CLT). The first part (Sections 1-9) is devoted to the total variation and the Kullback-Leibler distance (relative entropy). In the second part (Sections 10-15) we discuss general properties of Rényi and Tsallis divergences of order $α>1$, and then in the third part (Sections 16-21) we turn to the CLT and non-uniform local limit theorems with respect to these strong distances. In the fourth part (Sections 22-31), we discuss recent results on strictly subgaussian distributions and describe necessary and sufficient conditions which ensure the validity of the CLT with respect to the Rényi divergence of infinite order. |
| title | Rényi Divergences in Central Limit Theorems: Old and New |
| topic | Information Theory Probability 60E, 60F |
| url | https://arxiv.org/abs/2503.03926 |