I am trying to follow the proof given here: http://www.gautamkamath.com/writings/gaussian_max.pdf. In particular, I would like to understand the following crude bound mentioned in the paper:
My Attempt: \begin{align*} \mathbb{E}Y &= \mathbb{P}(|\{ i : X_i \geq 0 \}| \geq n/3) \mathbb{E}Y + \mathbb{P}(|\{ i : X_i \geq 0 \}| < n/3) \mathbb{E}Y \\ &\geq \mathbb{P}(|\{ i : X_i \geq 0 \}| \geq n/3) \mathbb{E}(\max_{1 \leq i \leq n/3} |X_i|) - \mathbb{P}(|\{ i : X_i \geq 0 \}| < n/3) \mathbb{E}|X_i|. \end{align*} I am not very sure if this is the right way to do it, but even if I understood the first inequality here correctly, how do we get to the second inequality mentioned? Here we are assuming $n \geq 2835$.