On the preservation of infinite divisibility under length-biasing

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.