Berry-Esseen bounds

Berry-Esseen bounds typically refer to any non-asymptotic bound on the deviation between a sum of random variables ${\sum }_i{X}_i$ and a target random variable $X_{\infty }$. They are often instrumental in proving central limit theorems and usually stated in terms of the KS distance, ${\rho }_{\text{KS}}$.

The classic Berry-Esseen bound, proved independently by both of them in 1945 is the following. Suppose $X_1,\dots ,X_n$ are independent with mean ${\mu }_i$, variance ${\sigma }_i^2=\mathbb{E}[(X_i-{\mu }_i{)}^2]$ and $B_n^2={\sum }_{i\le n}{\sigma }_i^2$. Then,

$${\rho }_{\text{KS}}\left(\frac{1}{B_n}\sum _{i\le n}(X_i-{\mu }_i),N(0,1)\right)\le \frac{C{\sum }_{i\le n}\mathbb{E}|X_i-{\mu }_i{|}^3}{(B_n^2{)}^{3/2}}.$$

for some universal constant $C$. Note that this assumes the existence of a third moment.

For iid observations, this reduces to

$${\rho }_{\text{KS}}\left(\frac{\sqrt{n}}{\sigma }({\overline{X}}_n-\mu ),N(0,1)\right)\le \frac{C\mathbb{E}|X-\mu {|}^3}{\sqrt{n}\sigma },$$

which is perhaps the more common form.

The best known constants $C$ are 0.5583 for independent rvs and 0.4690 for iid rvs. A lower bound (proved by Esseen is 0.4097). This thesis gives the optimal Berry-Esseen constant in the binomial case.

The convergence to 0 of the right hand side of the Berry-Esseen bound implies convergence of the Lindeberg function (Lindeberg-Feller CLT).