An optimal Berry-Esseen type theorem for integrals of smooth functions

We prove a Berry-Esseen type inequality for approximating expectations of sufficiently smooth functions $f$, like $f=|\cdot|^3$, with respect to standardized convolutions of laws $P_1,\ldots, P_n$ on the real line by corresponding expectations based on symmetric two-point laws $Q_1,\ldots,Q_n$ isoscedastic to the $P_i$. Equality is attained for every possible constellation of the Lipschitz constant $\|f"\|^{}_{\mathrm{L}}$ and the variances and the third centred absolute moments of the $P_i$. The error bound is strictly smaller than $\frac 16$ times the Lyapunov ratio times $\|f"\|^{}_{\mathrm{L}}$, and tends to zero also if $n$ is fixed and the third standardized absolute moments of the $P_i$ tend to one. In the homoscedastic case of equal variances of the $P_i$, and hence in particular in the i.i.d. case, the approximating law is a standardized symmetric binomial one. The inequality is strong enough to yield for some constellations, in particular in the i.i.d. case with $n$ large enough given the standardized third absolute moment of $P_1$, an improvement of a more classical and already optimal Berry-Esseen type inequality of Tyurin (2009). Auxiliary results presented include some inequalities either purely analytical or concerning Zolotarev's $\zeta$-metrics, and some binomial moment calculations.