Exercise 2.19: Maxima of Orlicz Variables

chapter 2

Denote $Z_n=\underset{i\in [n]}{max}\left|X_i\right|$. Use that $x\mapsto \psi ({\sigma }^{-1}x)$ is strictly increasing and convex: $$\begin{array}{}\text{(1)}& \psi ({\sigma }^{-1}\mathbb{E}[Z_n])\le \mathbb{E}[\psi ({\sigma }^{-1}{Z}_n)]\le \sum _{i=1}^n\mathbb{E}[\psi ({\sigma }^{-1}\left|X_i\right|)]\le n,\end{array}$$ so $$\begin{array}{}\text{(2)}& \mathbb{E}[Z_n]\le \sigma {\psi }^{-1}(n).\end{array}$$