I have quite a few questions regarding this paper, and in particular APPENDIX D, which contains the proof of propostion C.2:
With the following definitions in mind,
(denoting $\hat P \equiv \sum_{i=1}^n \delta_{X_i}/n$ the empirical measure). I believe there is a typo and that $\mu$ should be a $P$. Anyways, I am stuck on the first step of the proof, primarily because I'm a moron. It says this:
Here are the links to Maurer and Pontil (2010) and Gine and Nikl (2016), although truly (for me at least) only God knows where to find any of the arguments that correspond to this paper. How do you argue that $$\mathbb{E} \sup_{c, S} |\mathbb{E}_P f_{c,S} - \mathbb{E}_{\hat P} f_{c,S}| \leq \frac{\sqrt{2 \pi}}{n} \mathbb{E} \sup_{c, S} |\sum_{i=1}^n g_i f_{c, S}(X_i)| \quad \textbf{?}$$ I am genuinely and absolutely lost.
I understand the rest of the proof except for the above two inequalities and this next picture:
In particular the first and third inequalities make no sense. If at all possible, any help will be massively appreciated. I will gladly make a very large bounty for this if someone is able to help me understand what's going on.