  Let $Y_{1},Y_{2},\ldots $ be positive, nondegenerate, i.i.d. $G$ random
variables, and independently let $X_{1},X_{2},\ldots $ be i.i.d. $F$ random
variables. In this note we show that whenever $\sum X_{i}Y_{i}/\sum Y_{i}$
converges in distribution to nondegenerate limit for some $F\in \mathcal{F}$,
in a specified class of distributions $\mathcal{F}$, then $G$ necessarily
belongs to the domain of attraction of a stable law with index less than 1. The
class $\mathcal{F}$ contains those nondegenerate $X$ with a finite second
moment and those $X$ in the domain of attraction of a stable law with index
$1<\alpha <2$.
