The effect of random scale changes on limits of infinitesimal systems. (1978)

• Record Type: Journal Article
• Authors: Brockett, Patrick L.
• Is Part Of: International journal of mathematics and mathematical sciences. Volume 1:Number 3(1978)
• Page Start: 339
• Page End: 372
• Publication Date: 1978
• Subjects: general central limit theorem -- products of random variables in the domain of attraction of stable laws -- Lévy spectral function; Mathematics -- Periodicals; 510.5
• DOI: 10.1155/S0161171278000368
• Languages: English
• ISSNs: 0161-1712

Abstract

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.