Some Questions on the Berry-Esseen Theorem

Question

Berry-Esseen Theorem (Jacod - Protter):

Let $(X_j)_{j \geq 1}$ be i.i.d. and suppose that $E\{|X_j|^3\}<\infty$. Let $G_n(x) = P\left(\frac{S_n-n\mu}{\sigma \sqrt{n}}\leq x\right)$ where $\mu = E\{X_j\}$ and $\sigma^2 = \sigma^2_{X_j} < \infty$. Let $\Phi(x) = P(Z \leq x)$, where $L(Z) = N(0,1)$. Then $$\sup_{x\in\mathbb R}\left|G_n(x)-\Phi(x)\right| \leq c \frac{E\{|X_j|^3\}}{\sigma^3 \sqrt{n}},$$ where $c$ is a constant.

Help me to understand this theorem intuitively.

Some questions:

1. How is this related to the Classic Central Limit Theorem?
2. What does this $c$ represents?
3. I've searched and some says that "It is quantitative version of the CLT", what does it mean?
4. The book of Jacod-Protter hinted about the "rate of convergence", how does it relate to the Berry-Esseen Theorem, is there a relation of the "rate of convergence" and the constant $c$? If so, how are they related?

Thank you for your help.

Answer 1

1. The classical CLT says that $G_n(x)\to\Phi(x)$ for each $x$, which is implied by the Berry-Esseen inequality.
2. $c$ is a number, e.g. $7.59$. It has been optimized over the years. See the wikipedia article here.
3. Yes it gives an explicit uniform bound on the difference between the CDF $G_n$ and the standard normal CDF $\Phi$. The standard CLT only establish pointwise convergence (point 1).
4. The rate of convergence is related to $1/\sqrt{n}$. The Berry-Esseen inequality states that the convergence happens uniformly at speed at least $1/\sqrt{n}$.