SupposeS={{Xnj, j=1,2,…,kn}}is an infinitesimal system of random variables whose centered sums converge in law to a (necessarily infinitely divisible) distribution with Levy representation determined by the triple(γ,σ2,M). If{Yj, j=1,2,…}are independent indentically distributed random variables independent ofS, then the systemS′={{YjXnj,j=1,2,…,kn}}is obtained by randomizing the scale parameters inSaccording to the distribution ofY1. We give sufficient conditions on the distribution ofYin terms of an index of convergence ofS, to insure that centered sums fromS′be convergent. If such sums converge to a distribution determined by(γ′,(σ′)2,Λ), then the exact relationship between(γ,σ2,M)and(γ′,(σ′)2,Λ)is established. Also investigated is when limit distributions fromSandS′are of the same type, and conditions insuring products of random variables belong to the domain of attraction of a stable law.