Personally, I've always liked this name: Berry-Esseen. So sleek, so stylish. A certain Blackberry vibe to it. The theorem, however, owes its name to Andrew C. Berry and Carl-Gustav Esseen, both of whom independently proved (the first iteration of) the undermentioned result in the 1940s.
What is the Berry-Esseen Theorem about? It is a theorem that declares a (1) uniform bound on the extent of (2) "normality" of the sample mean (3) under CLT under the (4) existence of the 3rd order absolute moment. What do I mean? Let's go step by step:
Theorem: Berry-Essen Theorem (BET)
Setup: RVs $X_1, X_2, \dots$ i.i.d. have $\E X_i =0$, $\mathbb E X_i^2 = \sigma^2 >0$ and $\mathbb E |X_i|3 = \gamma < \infty$.
Notation: $\bar X_n := \frac{X_1 + \dots + X_n}{n}$. Let $F_n$ be the CDF of $\frac{\bar X_n}{\sigma/\sqrt n}$.
Existing: From CLT we already know $\frac{\bar X_n}{\sigma/\sqrt{n}} \to \mathcal N (0,1)$, i.e. $F_n \to \Phi$.
BET further states:
\[\sup_{x \in \RR} |F_n(x) - \Phi(x)| \leq \frac{C\gamma}{\sigma^3 \sqrt n}$\]
Explainer:
- The larger a sample, the more the "normality" in the distribution of its sample mean.
- BET is a bound on how much the distribution of $\sqrt n \bar X_n/ \sigma$ (standardized sample mean) can deviate from normality.
- The bound is supremal (on the worst possible deviation), and uniform (the bound works for $x$).
Note:
- Esseen (1956) proved that $C \geq \frac{\sqrt{10} + 3}{6 \sqrt{2 \pi}}$. Recent expositions have proved $C \leq 0.4748$. But nobody knows what $C$ exactly is.
- $F_n$ is the (small sample) CDF $\sqrt n \bar X/ \sigma$ based on $X_1, \dots, X_n$. It is not an empirical CDF, because of technically $F_n(t) = \E \left[\sum \mathbb I(X_i \le t)\right]$ is a deterministic function: the $\E$ makes it non-random.
What came after BET?