Is there an "elementary" proof that $\alpha$-stable random variables only exist for $0 < \alpha \le 2$? By elementary I mean without using Fourier transforms. I'd be happiest with either a direct probabilistic argument, or a geometric argument, e.g., referring to embeddings of $L_p$ spaces (as long as it doesn't depend on results which are themselves proved via Fourier analysis).

Note that what I'm interested in here is the fact that $\alpha$-stable random *don't* exist for $\alpha > 2$, not the fact that they *do* exist for $\alpha \le 2$.

**Edit:** To clarify, for the purposes of this question I define an $\alpha$-stable random variable as follows. Given $\alpha > 0$, $X$ is $\alpha$-stable if, whenever $X_1, \ldots, X_n$ are independent copies of $X$, there is a real number $d=d(n)$ such that $X_1 + \cdots + X_n$ has the same distribution as $n^{1/\alpha} X + d$. And I'm quite happy to simplify to the situation when $d=0$.