%0 Journal Article
%T The effect of random scale changes on limits of infinitesimal systems
%A Patrick L. Brockett
%J International Journal of Mathematics and Mathematical Sciences
%D 1978
%I Hindawi Publishing Corporation
%R 10.1155/s0161171278000368
%X Suppose S={{Xnj, ¡é ? ¡ë ¡é ? ¡ë ¡é ? ¡ëj=1,2, ¡é ? |,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, ¡é ? ¡ë ¡é ? ¡ë ¡é ? ¡ëj=1,2, ¡é ? |} are independent indentically distributed random variables independent of S, then the system S ¡é ? 2={{YjXnj,j=1,2, ¡é ? |,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 ¡é ? 2 be convergent. If such sums converge to a distribution determined by (   3 ¡é ? 2,(     ¡é ? 2)2,    ), then the exact relationship between (   3,    2,M) and (   3 ¡é ? 2,(     ¡é ? 2)2,    ) is established. Also investigated is when limit distributions from S and S ¡é ? 2 are of the same type, and conditions insuring products of random variables belong to the domain of attraction of a stable law.
%K general central limit theorem
%K products of random variables in the domain of attraction of stable laws
%K L¨¦vy spectral function.
%U http://dx.doi.org/10.1155/S0161171278000368