I am trying to show that the sub-Gaussian norm is a proper norm:
$$\|X\|_{\psi_2}=\text{inf}\bigg\{t>0:\mathbb{E}\bigg[\text{exp}\bigg(\frac{X^2}{t^2}\bigg)\bigg]\bigg\}$$
I was able to show the triangle inequality, and just want to make sure my logic on showing the homogeneity property is correct:
$$\|cX\|_{\psi_2}=\text{inf}\bigg\{t>0:\mathbb{E}\bigg[\text{exp}\bigg(\frac{c^2X^2}{t^2}\bigg)\bigg]\bigg\}$$
Def $k=\frac{t}{|c|}$ then
$$\|cX\|_{\psi_2}=\text{inf}\bigg\{|c|k>0:\mathbb{E}\bigg[\text{exp}\bigg(\frac{X^2}{k^2}\bigg)\bigg]\bigg\}=|c|\text{inf}\bigg\{k>0:\mathbb{E}\bigg[\text{exp}\bigg(\frac{X^2}{k^2}\bigg)\bigg]\bigg\}=|c|\|X\|_{\psi_2}$$
Is this correct?
Best Answer
You proof is correct, and basically uses the fact that $\inf_{a\in A} \lvert c\rvert a=\lvert c\rvert \inf_{a\in A} a$.