Exercise 2.19: Maxima of Orlicz Variables

chapter 2

Denote \(Z_n = \max_{i \in [n]}\abs{X_i}\). Use that \(x \mapsto \psi(\sigma^{-1} x)\) is strictly increasing and convex: \begin{equation} \psi(\sigma^{-1}\E[Z_n]) \le \E[\psi(\sigma^{-1}Z_n)] \le \sum_{i=1}^n \E[\psi(\sigma^{-1}\abs{X_i})] \le n, \end{equation} so \begin{equation} \E[Z_n] \le \sigma \psi^{-1}(n). \end{equation}