The effect of random scale changes on limits of infinitesimal systems

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.

Keywords

• general central limit theorem
• products of random variables in the domain of attraction of stable laws
• Lévy spectral function.