On the Preservation of Infinite Divisibility under Length-Biasing
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.
