Orlicz norm

• Definition
  • Let $\Psi$ be an increasing convex function from $[0, \infty)$ onto $[0, \infty)$ and $X$ be a random variable. The Orlicz norm $\| X \|_\Psi$ is defined by $$ \| X \|_\Psi = \text{inf} \left\{ c > 0 : E \Psi \left( \frac{|X|}{c} \right) \leq 1 \right\}, $$ where the infimum over the empty set is $\infty.$
• Examples of $\Psi$
  1. $\Psi(x) = \text{exp}(x^p) - 1 \equiv \Psi_p(x)$ for $p \geq 1,$ and in particular $\Psi_1$ and $\Psi_2$ corresponding to random variables which are “sub-exponential” or “sub-Gaussian” respectively.