On the Breiman conjecture

Abstract

Let $Y_1,Y_2,\dots$ be positive, nondegenerate, i.i.d. $G$ random variables, and independently let $X_1,X_2,\dots$ 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 $1<\alpha <2$.