Characterizations, length-biasing, and infinite divisibility

Abstract

Suppose L(X) is the law of a positive random variable X, and Z is positive and independent of X. Admissible solution pairs (L(X),L(Z)) are sought for the in-law equation X̂ ≅ X o Z, where L(X̂) is a weighted law constructed from L(X), and o is a binary operation which in some sense is increasing. The class of weights includes length biasing of arbitrary order. When o is addition and the weighting is ordinary length biasing, the class of admissible L(X) comprises the positive infinitely divisible laws. Examples are given subsuming all known specific cases. Some extensions for general order of length-biasing are discussed.