A Berry--Esseen Bound of Order $(1+\log n)/\sqrt n$ for Peng's Central Limit Theorem

Source abstract

We derive a quantitative central limit theorem under sublinear expectations from a spatial endpoint regularity estimate for the one-dimensional $G$-heat equation. For bounded Lipschitz initial data, the second spatial derivative and the time derivative of its solution are Lipschitz in space and $1/2$-Hölder in time away from the initial time. In particular, \([u_{xx}(t,\cdot)]_{\mathrm{Lip}}\le C_G\Lip(φ)/t\). Combining this estimate with the nonlinear Stein identity introduced in Song \cite{Song2020} and a replacement argument yields a bound of order $(1+\log n)/\sqrt n$ for bounded Lipschitz tests when the third absolute moment is finite and the lower variance is positive. The final replacement step is estimated directly, avoiding the nonintegrable singularity at time zero.