If $X_i$ are independently and identically distributed $N(0,\sigma^2)$ then $Y=\sum X_i \sim N(0,n\sigma^2)$, i.e. $\sum X_i \sim \sqrt{n}X_i$. That raises two questions:
Is a zero-mean normal distribution the only such example where the sum of a sample of $n$ iid random variables has the same distribution as the individual components, though rescaled by $\sqrt{n}$? For distributions with finite variance, the Central Limit Theorem suggests to me this should be the only possibility, but are there examples without a finite variance?
If the "independent" requirement is removed but identically distributed kept (and perhaps some form of exchangeability), there are other examples, including $Z_i=\frac{X_i}{\sum X_i^2}$ when $X_i \sim N(0,\sigma^2)$ iid (see a recent question). What other examples have this property? Is it possible to state what properties such distribution must have?
