Central Limit Theorem for Rényi Divergence of Infinite Order

Abstract

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 ${\sup }_x(p_n(x)-\phi (x))/\phi (x)\to 0$ as $n\to \infty$.

Corrected typos and updated section 11