Consider a standard statistical estimation problem with iid real observations $\{X_i\}_{i=1}^N$. For a collection of real functions $\mathcal{F}$, I want to get an estimate of the uniform rate of convergence
$E[\sup_{f\in\mathcal{F}} (\frac{1}{N}\sum_i f(X_i) - E[f(X)])^2]$
For a single square integrable $\mathcal{F}=\{f\}$, the above quantity is classically $O(N^{-1})$. Given $f$ bounded and equicontinuous, the results of Ranga Rao* show that $\sup_{f}\{\frac{1}{N}\sum_i f(X_i) - E[f(X)]\}\to 0$ almost surely, but don't provide a rate.
I can show, assuming $\mathcal{F}$ is a family of functions which are bounded with bounded derivatives (uniformly both in space and $\mathcal{F}$) and $P$ is a measure under which $X$ has a $C^3$ density, that an error of $O(N^{-2/5 })$ is guaranteed (this comes from using a kernel density estimator as an interim step).
It seems a better estimate, under less assumptions, should be possible - does anyone know a source?
*R. Ranga Rao, Relations between Weak and Uniform convergence of measures with applications, Ann. Math. Stat, 1962