The effect of random scale changes on limits of infinitesimal systems

Suppose S={{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 ( 3, 2,M). If {Yj, ￠ € ‰ ￠ € ‰ ￠ € ‰j=1,2, ￠ € |} are independent indentically distributed random variables independent of S, then the system S ￠ € 2={{YjXnj,j=1,2, ￠ € |,kn}} is obtained by randomizing the scale parameters in S according to the distribution of Y1. We give sufficient conditions on the distribution of Y in terms of an index of convergence of S, to insure that centered sums from S ￠ € 2 be convergent. If such sums converge to a distribution determined by ( 3 ￠ € 2,( ￠ € 2)2, ), then the exact relationship between ( 3, 2,M) and ( 3 ￠ € 2,( ￠ € 2)2, ) is established. Also investigated is when limit distributions from S and S ￠ € 2 are of the same type, and conditions insuring products of random variables belong to the domain of attraction of a stable law.