Long suggested by Shannon's EPI, it was finally proved by Artstein, Ball, Barthe and Naor in 2004 that entropy is monotone along the CLT.  In this talk, we revisit this result and give an elementary proof using a characterization of maximal correlation for iid sums due to Dembo, Kagan and Shepp.  A similar argument also yields strict decay of Stein discrepancy, which ultimately leads to dimension-free quantitative CLTs in the Wasserstein W2 distance, provided certain regularity conditions hold.  To our knowledge, these latter estimates for W2 are the first with both optimal convergence rate and dependence on dimension, holding for a large class of probability measures.

Joint work with Max Fathi (Institut de Mathématiques de Toulouse).