2.7.1 A More General View: Orlicz Spaces.
Orlicz funcution is a function $\phi : [0,\infty]\to[0,\infty]$ s.t. $\phi(0)=0$, $\phi(x)\to\infty$ as $x\to\infty$.
| Orlicz norm $\Vert .\Vert_\phi :=\inf{t>0:\mathbb{E}[\phi( | X | /t)]\le 1>}$. |
Orlicz space $\mathbf{L}\phi = \mathbf{L}\phi(\Omega,\Sigma,\mathbb{P})$ consists of all random variabesl X on probability space $(\Omega,\Sigma,\mathbb{P})$ s.t. $\mathbf{L}\phi := {X: \Vert X\Vert\phi<\infty}$.
Exercise 2.7.11. $\Vert X\Vert_\phi$ is norm on $\mathbf{L}\phi$. $\mathbf{L}\phi$ is complete, thus Banach space.
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Example 2.7.12. $L^p$ space. Orlicz function $\phi(x)=x^p, p\ge 1. Orlicz space $L_\phi$ is $L^p$ space.
Example 2.7.13. $L_{\phi_2}. Orlicz function $\phi_2(x) :=e^{x^2}-1$, then $\Vert X\Vert_{\phi_2} is exactly the sub-gaussian norm, and the Orlicz space $L_\phi_2$ contains all subgaussian-random variables.
Remark 2.7.14. $L^\infty\subset L_phi_2 \subset L^p, \forall p\in p[1,\infty)$. where $L^\infty$ the space of all bounded random variables.
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