Let $f:[0,1]^2\rightarrow\mathbb{R}$ and $D = \{x_1,\dots,x_n\}$ be some real independent random variables in $[0,1]$.
I need to bound:
$$ \mathbb{E}_D\left[\left(\sup_{x\in[0,1]}\frac{1}{n}\sum_{i=1}^nf(x,x_i)-E[f(x,x_i)]\right)^2\right] $$ Is there any chance to bound this expectation by something like $$ \frac{c}{n}\operatorname{Var}\left(\sup_{x\in[0,1]}f(x,x_1)\right). $$
Any suggestions are very appreciated! Thanks.