Berry–Esseen theorem

Remarks

The theorem gives a rate for the central limit theorem, which by itself only says that the left side tends to $0$. The classical proof compares characteristic functions and uses Esseen's smoothing inequality, which bounds the distance between two distribution functions by an integral of the difference of their characteristic functions. The best value of $C$ is not known; it is known that one may take $C < 1/2$.

The rate $n^{-1/2}$ cannot be improved in general. Let $X_k$ be $1$ or $-1$ with probability $1/2$ each, so $\sigma = \rho = 1$. For even $n$, the variable $S_n = X_1 + \cdots + X_n$ takes the value $0$ with probability $\binom{n}{n/2} 2^{-n}$, which is of order $n^{-1/2}$. So $F_n$ has a jump of that size at $0$, while $\Phi$ is continuous, and the supremum is at least half the jump.

Going deeper

For a student

The bound turns the normal approximation into a quantitative statement, which matters in statistics when the sample size is fixed. It needs a finite third moment; with only a finite variance, the convergence in the central limit theorem can be arbitrarily slow. There are versions for independent variables that are not identically distributed, with $\sum_k E[\abs{X_k}^3] / s_n^3$ in place of $\rho / (\sigma^3 \sqrt{n})$, and multivariate versions. Stein's method gives another proof and extends to dependent variables.