The effect of random scale changes on limits of infinitesimal systems

Patrick L. Brockett

International Journal of Mathematics and Mathematical Sciences, 1978, vol. 1, 1-34

Abstract:

Suppose S = { { 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 of S , then the system S ′ = { { Y j X n j , j = 1 , 2 , … , k n } } is obtained by randomizing the scale parameters in S according to the distribution of Y 1 . 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 , σ 2 , 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.

Date: 1978
References: Add references at CitEc
Citations:

Downloads: (external link)
http://downloads.hindawi.com/journals/IJMMS/1/679789.pdf (application/pdf)
http://downloads.hindawi.com/journals/IJMMS/1/679789.xml (text/xml)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:679789

DOI: 10.1155/S0161171278000368

Access Statistics for this article

More articles in International Journal of Mathematics and Mathematical Sciences from Hindawi
Bibliographic data for series maintained by Mohamed Abdelhakeem ().