Let $(X_n)_{n}$ be a sequence of i.i.d. real random variables with mean $\mu \in \mathbb{R}$ and variance $\sigma^2 > 0$. Let $S_n = X_1 + ... + X_n$.
The usual central limit theorem ensures that $\sqrt{n}(S_n/n - \mu)$ converges in distribution to $\mathcal{N}(0,\sigma^2)$. This is often used to motivate the heuristic approximation that $S_n/n$ approximatively follows a $\mathcal{N}(\mu,\sigma^2/n)$ distribution.
Is there a way of using the central limit theorem (or something else) to actually control how close $S_n/n$ is to $\mathcal{N}(\mu,\sigma^2/n)$?
For instance, would it be possible to assess the convergence speed of $|F_n - G_n| $ to zero, where $F_n$ is the cumulative distribution function of $S_n/n$, and $G_n$ is the one of $\mathcal{N}(\mu,\sigma^2/n)$?