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 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.