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We explore asymptotically optimal bounds for deviations of distributions of independent Bernoulli random variables from the Poisson limit in terms of the Shannon relative entropy and Rényi/relative Tsallis distances (including Pearson’s χ2). This part generalizes the results obtained in Part I and removes any constraints on the parameters of the Bernoulli distributions.
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This content will become publicly available on January 1, 2026
Rényi divergences in central limit theorems: Old and new
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 Tsall is divergences of order alpha > 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.
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- Award ID(s):
- 2154001
- PAR ID:
- 10613524
- Publisher / Repository:
- Inst. Math. Statist.
- Date Published:
- Journal Name:
- Probability Surveys
- Volume:
- 22
- ISSN:
- 1549-5787
- Page Range / eLocation ID:
- 1–75
- Subject(s) / Keyword(s):
- Central limit theorem, Rényi divergence
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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