I am trying to understand the following assertion made in a paper (without any explanation as such). Let $\{g_i \}_{i=1}^{d} $ be an i.i.d. collection of standard Gaussian random variables, and let $||\cdot \||$ be an arbitrary norm on $\mathbb{R}^d$. Then the following assertion is made:
Claim:
$\mathbb{E}(\left( \lVert\sum_{i=1}^{d}g_ie_i \rVert \right)^2 \leq C \left(\sum_{i=1}^{d}\lVert e_i \rVert ^2 \right)$, although I suspect the stronger assertion that $\mathbb{E}(\left( \lVert\sum_{i=1}^{d}g_ie_i \rVert^2 \right) \leq C \left(\sum_{i=1}^{d}\lVert e_i \rVert ^2 \right)$ may also be true.(Here $\{e_i\}$ is the canonical basis on $\mathbb{R}^d$).
Where the constant $C$ does not depend on $d$ (but can possibly depend on the norm $\lVert \cdot \rVert$). I am not able to understand why this is true. My question is: is there a simple explanation for this fact?
A few remarks or observations are:
This looks quite a lot like Khintchine’s inequality, however the difference is that the norm $\lVert \cdot \rVert$ here is not necessarily an $\ell^p$ norm, and also the random variables here are Gaussian as opposed to Rademacher random variables. As a consequence, I am not able to use the adapt the usual proof of Khintchine's inequality to this setting.
Searching through the internet makes me think that this is related the the concept of the type and co-type of a Banach space, but I'm unfamiliar with this topic and haven't found exactly what I want. Any commentary on how this is related to this concept will be useful.