I'm trying to understand [sub-gaussian RVs][1] to see if they could be relevant to my work.

The common definition of a sub-gaussian RV is the following. X is $\sigma$ sub-gaussian if its laplace transform / moment generating function is smaller than that of a Gaussian RV of standard deviation $\sigma$

$$ E(\exp(tX)) \leq \exp(\sigma^2 t^2 / 2) $$

Note that this characterizes

Another characterization of sub-gaussian variables is:

$$ \exists a, E(\exp(a X^2)) \leq 2 $$

And it seems to me that (almost) all random variables check that condition. Indeed if we look at the function (which is a form of moment generating function):

$$ a \rightarrow f(a) = E(\exp(a X^2)) $$

then we know the value at 0: $f(0)=1$ and, if f is continuous, then we can find a value of $a$ that checks the condition.

Does that mean that pretty much everyone (unless $f$ is absurdly miss-behaved) is sub-gaussian ? [1]: http://www.math.ualberta.ca/~orivasplata/publications/subgaussians.pdf