how to estimate the detailed convergence rate in helly's theorem (probability)? We all know one of the versions of Helly's theorem in probability course (c.f. the book 'Approximation Theorems of Mathematical Statistics', page 352. See as follows):
Let $f(x)$ be the continuous function, and let the sequences of the non-decreasing uniformly bounded functions: $F_1(x), F_2(x), ... , F_n(x),...$ converge weakly to the function $F(x)$ on some finite interval $a\leq x \leq b$. Then
$\lim\limits_{n\rightarrow +\infty}\int_{a}^b f(x) d F_n (x) = \int_{a}^b f(x) d F (x)$.
My question is: How can we estimate the detailed convergence rate of the above convergence rate, i.e., w.r.t the sample $n$? Do we have any corresponding tools? Thanks, everyone!
 A: Without knowing the context, it's probably difficult to say anything general about the convergence rate. But 
a)
$|\int_{a}^{b}f(x)dF_{n}(x)-\int_{a}^{b}f(x)dF(x)|\\
\leq \sup|f(x)|*||\mu_{n}-\mu||_{TV}$, 
where $||\mu_{n}-\mu||_{TV}$ is the total variation distance between
the measures corresponding to the distribution functions. In specific
situations, the total variation distance can be bounded.
b) if $f$ is smooth and compactly supported and let $supp(f)\subset (a,b)$, then an integration by parts shows that
$|\int_{a}^{b}f(x)dF_{n}(x)-\int_{a}^{b}f(x)dF(x)|\\
=|\int_{a}^{b}f^{'}(x)F_{n}(x)dx-\int_{a}^{b}f'(x)F(x)dx|\\
\leq \sup|f^{'}(x)|*\sup_{x\in[a,b]}|F_{n}(x)-F(x)|*(b-a)\\
\leq \sup|f^{'}(x)|*\sup|F_{n}(x)-F(x)|*(b-a)$
$\sup|F_{n}(x)-F(x)|$ is called the Kolmogrov-Smirnov distance and in specific situations, it can be bounded.
c) Under the same assumptions in (b) and assume the measures corresponding to the distribution function are probabability measures. Then
$|\int_{a}^{b}f(x)dF_{n}(x)-\int_{a}^{b}f(x)dF(x)|\\
= |E[f(X_{n})] - E[f(X)]|\\
\leq \sup |f^{'}(x)| E[|X_{n} - X|]
$
where in the last step, I constructed a "coupling" between $X_{n}$ and $X$ (a joint distribution having the correct marginals). Certainly, I can take $\inf$ over all couplings to make the bound tight and that will give
$|\int_{a}^{b}f(x)dF_{n}(x)-\int_{a}^{b}f(x)dF(x)|\\
\leq \sup |f^{'}(x)| \inf_{\text{all coupling of $X_{n}$ and X}}E[|X_{n} - X|]\\
=  \sup |f^{'}(x)|*Wasserstein-distance(X_{n},X)
$
The wassertein-distance can also be bounded in many cases
d) Now let's consider the setting in the central limit theorem where $F_n$ is the distribution function of the rescaled sums and $F$ the distribution function of the standard normal. Then the Berry-Esseen theorem gives precisely a bound of the Kolmogrov-Sirnov distance as $O(1/{\sqrt{n}})$ (checkout the wiki). In fact, the total variation distance can also be bounded, checkout the papers by Sergey Bobkov.
