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

Abstract

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 $P_1,\dots ,P_n$ on the real line by corresponding expectations based on symmetric two-point laws $Q_1,\dots ,Q_n$ isoscedastic to the $P_i$. Equality is attained for every possible constellation of the Lipschitz constant $\Vert f"{\Vert }_{\mathrm{L}}^{}$ and the variances and the third centred absolute moments of the $P_i$. The error bound is strictly smaller than $\frac{1}{6}$ times the Lyapunov ratio times $\Vert f"{\Vert }_{\mathrm{L}}^{}$, 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.

39 pages, 1 figure, 86 references. As compared to the first version, the introductory section has been changed