$p$-stability singles out $p = 0.5, 1, 2$. Specifically, there is no probability distribution $P$ such that the linear combination $\sum^n a_i X_i$ is distributed as $\|a\|_p Y$, where $X_1 ... X_n$ and $Y$ are random variables distributed according to $P$, if $p$ is not $1/2, 1, 2$.
Suresh Venkat
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