Weighted sum of identical distributed random variables Suppose $X_1$, $X_2$, $\ldots$ ,$X_N$ are identically distributed (not necessarily independent). 
Then, given $a_1+a_2+\ldots+a_N=1$, and let $S=a_1 X_1 + a_2 X_2 + \ldots + a_N X_N$. Does $S$ follow the same distribution as $X$?

Not necessarily. Fix an arbitrary random variable $X$ whose law is not a Dirac mass at $0$ (say standard normal, for example) and set $X_1=X$ and $X_2=-X$. $S=\frac{1}{2}X_1 + \frac{1}{2}X_2 = 0$ does not have the same law as $X$.
