Entropy inequalities and the Central Limit Theorem

Motivated by Barren (1986, Ann. Probab. 14, 336-342), Brown (1982, Statisti cs and Probability: Essays in Honour of C.R. Rao, pp. 141-148) and Carlen a nd 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 i n the relative entropy sense of Kullback-Leibler, which is strictly stronge r than L-1. In the IID case we recover the main result of Barren [1] (c) 20 00 Elsevier Science B.V. All rights reserved.