The Berry-Esseen Bound

The Central Limit Theorem is a statement about a limit: as sample size $n$ goes to infinity, the distribution of a standardized sample mean approaches the normal distribution exactly. It's silent on the question every practitioner actually cares about — at $n=30$, or $n=100$, how close to normal is it right now? The Berry-Esseen bound answers that by putting a hard number on the maximum possible gap between the true (unknown) distribution of the standardized mean and the normal distribution, for any finite $n$.

Concretely, if $F_n$ is the cumulative distribution function of the standardized sample mean of $n$ i.i.d. observations with finite third absolute moment $\rho$, and $\Phi$ is the standard normal CDF, then

$$\sup_x |F_n(x) - \Phi(x)| \le \frac{C \rho}{\sigma^3 \sqrt{n}}$$

for a universal constant $C$. In plain English: the biggest possible gap between the real distribution and the normal approximation shrinks at a rate of $1/\sqrt{n}$, and how large that gap starts out depends on $\rho/\sigma^3$ — a measure of how skewed or heavy-tailed the underlying data is. Data that's already close to symmetric and thin-tailed converges to normal fast; data with a lot of skew or occasional extreme values needs a much larger $n$ before the normal approximation is trustworthy.

This is why "$n>30$ is enough for the CLT" is a rule of thumb and not a law: the Berry-Esseen bound shows explicitly that the sample size needed depends on the shape of the underlying distribution, and daily financial returns — skewed, fat-tailed — sit exactly in the case where that rule of thumb is least reliable.