I was reading the book
Technical Incerto - Statistical Consequences of Fat Tails
by Nassim Nicholas Taleb, in this book the autor writes the following (page 28):
"Membership in the subexponential class does not satisfy the so-called Cramer condition, allowing insurability, as we illustrated in Figure 3.1, recall out thought experiment in the beginning of the chapter. More technically, the Cramer condition means that the expectation of the exponential of the random variable exists."
On a foot note the autor leaves a very simple definition of the Cramer Condition, this condition is referenced along the book:
Let $X$ be a random variable. The Cramer condition: for all $r > 0$, $\mathbb{E}(e^{rX}) < + \infty$, where $\mathbb{E}$ is the expectation operator.
I am looking for a more formal definition of the Cramer Condition this author talks about, because I found this definition with some lack of formality, or maybe I am wrong and is more simple than I think it is and this will do. I am aware of subexponential distributions but I can't find anything related to this Cramer condition I thinks it as to do with large deviations theory. Am I in the right way? Will this definition suffice if I talk in a mathematically formal point?