International Journal of Mathematics and Mathematical Sciences
Volume 1 (1978), Issue 3, Pages 339-372

The effect of random scale changes on limits of infinitesimal systems

Patrick L. Brockett

Department of Mathematics, The University of Texas, Austin 78712, Texas, USA

Received 25 August 1977; Revised 3 April 1978

Copyright © 1978 Patrick L. Brockett. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 (γ,σ2,M). If {Yj,j=1,2,} are independent indentically distributed random variables independent of S, then the system S={{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 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 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.