For a random variable $X$, define
$$\lVert X\rVert_{\psi_2} =\inf \{k>0\mid \mathbb{E}[\exp((X/k)^2)]\leq 2\}$$
and for a random vector $\vec X$, define
$$\lVert \vec X\rVert_{\psi_2} = \sup_{\theta : \lVert \theta\rVert_2 = 1} \lVert \langle \vec X, \theta\rangle\rVert_{\psi_2}.$$
I'm interested in conditions which imply that $\lVert X\rVert_{\psi_2}$ (or $\lVert \vec X\rVert_{\psi_2}$) are large. I am in particular interested in quantitative/"fine-grained" results, e.g. statements such as
If (some condition depending on $K$ holds) then $\lVert X\rVert_{\psi_2} \geq f(K)$ for some function $f$
Examples of such conditions that I am aware of essentially boil down to converses for the standard equivalences between various definitions for sub-gaussianity (see Vershynin prop 2.5.2). For example
The moments $\mathbb{E}[|X|^p]^{1/p} \geq K\sqrt{p}$ for all $p\geq 1$,
If $\mathbb{E}[X] = 0$, then for all $\lambda>0$ $\mathbb{E}[\exp(\lambda X)] \geq \exp(K^2 \lambda^2)$
One example I know of that does not follow this naive trend is Exercise 3.4.5 of Vershynin (also located here), namely that if $\lVert \vec X\rVert_{\psi_2} =O(1)$ is constant independently of the dimension of $\vec X$, and $\vec X$ is supported on a finite set, then that set must have exponentially large cardinality.
Are there other results which are useful for showing a random variable $X$ does not have small sub-Gaussian parameter?
