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
Suppose 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 . If are independent indentically distributed random variables independent of , then the system is obtained by randomizing the scale parameters in according to the distribution of . We give sufficient conditions on the distribution of in terms of an index of convergence of , to insure that centered sums from be convergent. If such sums converge to a distribution determined by , then the exact relationship between and is established. Also investigated is when limit distributions from and are of the same type, and conditions insuring products of random variables belong to the domain of attraction of a stable law.
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