Orlicz norm

For a strictly increasing function $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, the $\psi$-Orlicz norm is

$${\left|X\right|}_{\psi }:=\inf \{u>0|\mathbb{E}\psi \left(\frac{|X|}{u}\right)\le 1\}.$$

The $\psi$-Orlicz norm is a norm on the space of those random variables $X$ such that ${\left|X\right|}_{\psi }$ is finite. Such a space is called an Orlicz space. It is a linear space.

Note that the Orlicz norm is a functional of the distribution, not a function of the random variable itself.

Relationship to $L^p$ spaces

Orlicz norms/spaces generalize Lp norms and spaces. This can be seen by taking $\psi (a)=a^p$ in which case

$${\mathbb{E}}_P\psi (|X|/u)=\int \frac{|X{|}^p}{u^p}dP.$$

This attains one iff $u=\int |X{|}^{p}dP$ meaning that ${\left|X\right|}_{\psi }={\left|X\right|}_p=\int |X{|}^{p}dP$.

Defining sub-Gaussian and sub-exponential random variables

Orlicz norms also provide a nice, unified way of dealing with sub-Gaussian sub-Gaussian distributions and sub-exponential distributions.

In particular, if we take $\psi (x)={\psi }_2(x)=\exp (x^2)-1$, then ${\left|\cdot \right|}_{{\psi }_2}$ is called the sub-Gaussian norm. A random variable is sub-Gaussian iff it is an element of the ${\psi }_2$-Orlicz space, i.e., if ${\left|X\right|}_{{\psi }_2}<\infty$. Note that

$${\left|X\right|}_{{\psi }_2}=\inf \{u>0|\mathbb{E}\exp (|X{|}^2/u^2)\le 2\}.$$

Meanwhile, if we take $\psi (x)={\psi }_1(x)=\exp (x)-1$, then ${\left|\cdot \right|}_{{\psi }_1}$ is the sub-exponential norm. A random variable is sub-exponential iff it is an element of the ${\psi }_1$-Orlicz space. Here

$${\left|X\right|}_{{\psi }_1}=\inf \{u>0|\mathbb{E}\exp (|X|/u)\le 2\}.$$