Linear combination of i.i.d. $Z_i$ distributed as $Z_1$

Question

A classical property of the Gaussian distribution is that, if $\{Z_i\}_{1 \leq i \leq n}$ are i.i.d. standardised Gaussian distributions (i.e. $Z_i \sim N(0,1)$) and $S = \sum_{i=1}^n a_i Z_i$ where $a_i \in \mathbb{R}$, then the law of $S$ is the same as the law of $\big( \sum a_i^2 \big)^{1/2} Z_1$.

Given a norm $\| \cdot \|$ on $\mathbb{R}^n$ which is invariant under permutation of the coordinates (e.g. $\ell^p$ or some Orlicz norm), can one find a (standardised) distribution such that if the $\{Z_i\}_{1 \leq i \leq n}$ are i.i.d then $\sum_{i=1}^n a_i Z_i = \| a\| Z_1$ in distribution?

[With my apologies if this is well-known.]

Answer 1

The distributions you're looking for are stable distributions. Basically, the only such norms you can take are $\ell^p$ norms for $1 \le p \le 2$.

If you don't need an honest norm, you can also take the $\ell^p$-"norm" (really quasinorm) for $0 < p < 1$. But, up to some simple transformations, those are actually the only functions of $a$ you can get here.