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