Can I always decompose a random variable in sum of iid random variables?

Question

Let $Z$ be a random variable. Can I always find a number $n \in \mathbb N > 1$, weights $w_i \neq 0$, and iid random variables $X_i$ such that

$$Z = w_1X_1 + \dots + w_n X_n$$?

Conversely, if I have a certain combination of $w_i$ and $X_i$, can I always choose the distribution that the $X_i$ follow so to make $Z$ follow whatever distribution I like?

Thanks!

(Maybe I should ask the converse in another question?)

Answer 1

No. For example this won't work for $P(Z=a)=P(Z=b)=1/2$, $a\not= b$. Since the convolution can only make measures less singular, $X_1$ must have purely discrete distribution also, but then it's easy to see that $\sum w_j X_j$ simply takes too many values (with positive probability).