An Edgeworth-type expansion is established for the entropy distance to the class of normal distributions of sums of i.i.d. random variables or vectors, satisfying minimal moment conditions.
"Rate of convergence and Edgeworth-type expansion in the entropic central limit theorem." Ann. Probab. 41 (4) 2479 - 2512, July 2013. https://doi.org/10.1214/12-AOP780