On the Breiman conjecture

Let Y1,Y2,…Y_{1},Y_{2},\ldots be positive, nondegenerate, i.i.d. GG random variables, and independently let X1,X2,…X_{1},X_{2},\ldots be i.i.d. FF random variables. In this note we show that for F∈FF\in \mathcal {F} in a specified class of distributions F\mathcal {F}, whenever ∑XiYi/∑Yi\sum X_{i}Y_{i}/\sum Y_{i} converges in distribution to a nondegenerate limit then G necessarily belongs to the domain of attraction of a stable law with index less than 1. The class F\mathcal {F} contains those nondegenerate XX with a finite second moment and those XX in the domain of attraction of a stable law with index 1>α>21>\alpha >2.