The effect of random scale changes on limits of infinitesimal systems. (1978) - Brockett, Patrick L.
Abstract : SupposeS = { { X n j,       j = 1, 2, …, k n } } 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, σ 2, M ) . If{ Y j,       j = 1, 2, … } are independent indentically distributed random variables independent ofS, then the systemS ′ = { { Y j X n j, j = 1, 2, …, k n } } is obtained by randomizing the scale parameters inS according to the distribution ofY 1 . We give sufficient conditions on the distribution ofY in terms of an index of convergence ofS, to insure that centered sums fromS ′ be convergent. If such sums converge to a distribution determined by( γ ′, ( σ ′ ) 2, Λ ), then the exact relationship between( γ, σ 2, M ) and( γ ′, ( σ ′ ) 2, Λ ) is established. Also investigated is when limit distributions fromS andS ′ are of the same type, and conditions insuring products of random variables belong to the domain of attraction of a stable law.