I'm preparing a lecture on stable distributions, and I'm trying to find a simple explanation of the following fact.
Suppose we are trying to come up with stable distributions. From the definition, it's clear that a distribution is stable iff its characteristic function $\phi$ satisfies $\phi(t)^n = e^{i t b_n} \phi(a_n t)$. The normal distribution, with chf $\phi(t) = e^{-t^2/2}$ clearly satisfies this with $b_n = 0$, $a_n = \sqrt{n}$. This suggests that we look for distributions with chfs of the form $\phi(t) = e^{-c |t|^\alpha}$. For $0 \le \alpha \le 2$, this is indeed a chf, and there is a nice proof in Durrett's book, constructing it as a weak limit using Lévy's continuity theorem. But:
For $\alpha > 2$, is there a simple reason why $\phi(t) = e^{-c |t|^\alpha}$ cannot be a chf?
Breiman's Probability proves a general formula for the chf of a stable distribution, using a representation formula for infinitely divisible distributions, but it's more work than I want to do for this.