On the Preservation of Infinite Divisibility under Length-Biasing. (20th July 2014)

• Record Type: Journal Article
• Authors: Pakes, Anthony G.
• Other Names: Eryílmaz Serkan Academic Editor.
• Is Part Of: Journal of probability. Volume 2014(2014)
• Publication Date: 2014-07-20
• Subjects: Probabilities -- Periodicals; Mathematical statistics -- Periodicals; Mathematical statistics; Probabilities; Periodicals; 519.2
• DOI: 10.1155/2014/703697
• Languages: English
• ISSNs: 2356-7589

Abstract

Abstract : The lawL ( X ) ofX ≥ 0 has distribution functionF and first moment0 < m < ∞ . The lawL ( X ^ ) of the length-biased version ofX has by definition the distribution functionm - 1 ∫ 0 x ‍ y d F ( y ) . It is known thatL ( X ) is infinitely divisible if and only ifX ^ = d X + Z, whereZ is independent ofX . Here we assume this relation and ask whetherL ( Z ) orL ( X ^ ) is infinitely divisible. Examples show that both, neither, or exactly one of the components of the pair( L ( X ), L ( X ^ ) ) can be infinitely divisible. Some general algorithms facilitate exploring the general question. It is shown that length-biasing up to the fourth order preserves infinite divisibility whenL ( X ) has a certain compound Poisson law or the Lambert law. It is conjectured for these examples that this extends to all orders of length-biasing.