Improved Bounds for the Entropic Central Limit Theorem in the Log-Concave Case

Let $\mu$ be a log-concave isotropic probability measure on $R^d$ , and let ${\mu }^n$ denote the law of the normalized sum of $n$ independent random vectors with law $\mu$. We prove that $$H({\mu }^n\mathrm{\mid }\gamma )\le \frac{C}{n}\min \{1+\log (ed{)}^{2}H(\mu \mathrm{\mid }\gamma ),d\}\mathrm{.}$$ Compared to the previous best known bound, our estimate replaces the polynomial dependence on the dimension by a polylogarithmic one. The proof relies on a new combination of the De Bruijn identity with Eldan’s stochastic localization process.