On the Preservation of Infinite Divisibility under Length-Biasing

Anthony G Pakes

doi:10.1155/2014/703697

Abstract

The law L(X) of X≥0 has distribution function F and first moment 0<m<∞. The law L(X^) of the length-biased version of X has by definition the distribution function m-1∫0x‍ydF(y). It is known that L(X) is infinitely divisible if and only if X^=dX+Z, where Z is independent of X. Here we assume this relation and ask whether L(Z) or L(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 when L(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.

Highlights

Let X be a nonnegative random variable whose distribution function (DF) and Laplace-Stieltjes transform (LST) is denoted by F(x) and φ(s), respectively
We extend this notation by writing Xr = LrX for a random variable whose DF is Fr and we denote the corresponding LST by φr(s)
Infdiv laws L(X) having a Levy exponent in TB comprise those laws which are the weak limits of sums of independent gamma distributed random variables, and they are called generalized gamma convolutions (GGCs)

Summary

Introduction

Let X be a nonnegative random variable whose distribution function (DF) and Laplace-Stieltjes transform (LST) is denoted by F(x) and φ(s), respectively. F = F1 and Lr denotes the length-bias operator of order r acting on distribution functions having a finite moment of order r We extend this notation by writing Xr = LrX for a random variable whose DF is Fr and we denote the corresponding LST by φr(s). Our algebraic manipulations become more involved as n increases, suggesting other methods are needed to resolve our conjectures one way or the other We end this introduction by reminding the reader that length-biasing is important in contexts such as random sampling of regions in spatial problems, for example, the distribution of the volume or surface area of spheres given the distribution of radii; see page 170 in [5]. The first of this pair of references cites other occurrences of length-biasing

Classes of Infdiv Laws

General Observations

A Compound Poisson Law

The Lambert Law