The effect of random scale changes on limits of infinitesimal systems

Suppose S = { { X n j , j = 1 , 2 , … , k n } } 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 , σ 2 , M ) . If { Y j , j = 1 , 2 , … } are independent indentically distributed random variables independent of S , then the system S ′ = { { Y j X n j , j = 1 , 2 , … , k n } } is obtained by randomizing the scale parameters in S according to the distribution of Y 1 . 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 ′ be convergent. If such sums converge to a distribution determined by ( γ ′ , ( σ ′ ) 2 , Λ ) , then the exact relationship between ( γ , σ 2 , M ) and ( γ ′ , ( σ ′ ) 2 , Λ ) is established. Also investigated is when limit distributions from S and S ′ are of the same type, and conditions insuring products of random variables belong to the domain of attraction of a stable law.