International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 1978 / Article

Open Access

Volume 1 |Article ID 679789 | https://doi.org/10.1155/S0161171278000368

Patrick L. Brockett, "The effect of random scale changes on limits of infinitesimal systems", International Journal of Mathematics and Mathematical Sciences, vol. 1, Article ID 679789, 34 pages, 1978. https://doi.org/10.1155/S0161171278000368

The effect of random scale changes on limits of infinitesimal systems

Received25 Aug 1977
Revised03 Apr 1978

Abstract

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.

Copyright © 1978 Hindawi Publishing Corporation. 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.


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