So the problem I'm having trouble with is part (a) of exercise 8.4.9 in Vershynin's High Dimensional Probability book.
The setup is as follows. Let us define a family of functions $\mathcal{F}$ and a target function $T$ as follows:
$$\mathcal{F}=\{ f:[0,1]\rightarrow \mathbb{R}, ||f||_\text{Lip} \leq L\}$$
and a target function $T:[0,1]\rightarrow [0,1]$.
Let $X, X_1, \cdots, X_n$ be iid. Show that the random process
$$X_f = \frac{1}{n}\sum_{i=1}^n [(f(X_i)-T(X_i))^2-E(f(X)-T(X))^2]$$
where $f\in \mathcal{F}$, has subgussian increments:
$$ ||X_f-X_g||_{\psi_2} \leq \frac{CL}{\sqrt{n}}||f-g||_\infty $$
$C$ is an absolute constant.
For some reason I am having a really hard time trying to show this. I have already tried the usual tricks such as rewriting the expectation in terms of another iid variable $Y_i$ so that you can write
$$X_f = \frac{1}{n}\sum_{i=1}^n E[(f(X_i)-T(X_i))^2-(f(Y_i)-T(Y_i))^2]$$
And tried manipulating this expression around, but I always got stuck trying to bound
$$||f(X_i)^2-g(X_i)^2 - (f(Y_i)^2-g(Y_i)^2)||^2_{\psi_2}$$
But I couldn't get $L$ and $||f-g||_\infty$ terms to come out simultaneously.