Entropy inequalities and the Central Limit Theorem

Motivated by Barron (1986, Ann. Probab. 14, 336–342), Brown (1982, Statistics and Probability: Essays in Honour of C.R. Rao, pp. 141–148) and Carlen and Soffer (1991, Comm. Math. Phys. 140, 339–371), we prove a version of the Lindeberg–Feller Theorem, showing normal convergence of the normalised sum of independent, not necessarily identically distributed random variables, under standard conditions. We give a sufficient condition for convergence in the relative entropy sense of Kullback–Leibler, which is strictly stronger than L1. In the IID case we recover the main result of Barron [1].