Entropic Central Limit Theorem for Rényi Entropy

We establish a central limit theorem for Rényi entropies when the Rényi parameters belong to (0, 1) for a large class of random vectors. This complements a celebrated result of Barron (1986). As an application, we show that a general Rényi entropy power inequality fails when the Rényi parameter is in (0, 1).