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 $1<\alpha <2$.
Categories: math.PR