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Central Limit Theorem for Rényi Divergence of Infinite Order
Sergey G. Bobkov, Friedrich Götze · 2024-02-04 · via math.PR updates on arXiv.org

For normalized sums $Z_n$ of i.i.d. random variables, we explore necessary and sufficient conditions which guarantee the normal approximation with respect to the Rényi divergence of infinite order. In terms of densities $p_n$ of $Z_n$, this is a strengthened variant of the local limit theorem taking the form $\sup_x (p_n(x) - \varphi(x))/\varphi(x) \rightarrow 0$ as $n \rightarrow \infty$.

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