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

Source abstract

Let $μ$ be a log-concave isotropic probability measure on $R^d$ , and let $μ^n$ denote the law of the normalized sum of $n$ independent random vectors with law $μ$. We prove that $$H(μ^n|γ)\leq \frac{C}{n} \min\left\{1+\log(ed)^2H(μ|γ)\ ,\ d\right\}.$$ 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.