Table of Contents
Journal of Probability
Volume 2014, Article ID 703697, 9 pages
http://dx.doi.org/10.1155/2014/703697
Research Article

On the Preservation of Infinite Divisibility under Length-Biasing

School of Mathematics & Statistics, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

Received 20 February 2014; Accepted 23 June 2014; Published 21 July 2014

Academic Editor: Serkan Eryílmaz

Copyright © 2014 Anthony G. Pakes. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The law of has distribution function and first moment . The law of the length-biased version of has by definition the distribution function . It is known that is infinitely divisible if and only if , where is independent of . Here we assume this relation and ask whether or is infinitely divisible. Examples show that both, neither, or exactly one of the components of the pair 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 when 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.