I am trying to show two things:
Firstly:
Suppose $X$ is a sub gaussian random variable, then we have: $$\|X\|_{L^p}:=\mathbb{E}[|X|^p]^{\frac{1}{p}}\le{C\|X\|_{\psi^2}\sqrt{p}},\space{}C>0$$
I don't really know where to even begin with this, I looked at the definition of the expectation here: $$\mathbb{E}[X]=\int_0^\infty{\mathbb{P}(X>t)}dt$$ But couldn't get anywhere.
And Secondly:
If $X=(X_1,...,X_N)$ is a random vector with independent subgaussian co-ordinates, are $\|X\|_2$ and $\|X\|_2-\sqrt{n}$ subgaussian?