Rényi Divergences in Central Limit Theorems: Old and New

2025/03/05 by Bobkov, S. G., Chistyakov, G. P., Götze, F.
#60E #60F #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2503.03926

Abstract

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.

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• Rényi divergence and the central limit theorem
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• Central Limit Theorem for Rényi Divergence of Infinite Order
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