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. 1. Introduction Let be a nonnegative random variable whose distribution function (DF) and Laplace-Stieltjes transform (LST) is denoted by and , respectively. If the law of , , is infinitely divisible (infdiv), then , where the Laplace exponent (or cumulant function) is a Bernstein function, denoted by . This means that there is a measure (called the Lévy measure) on satisfying and a constant such that In this case is the left-extremity of the support of and hence we will set , thereby losing no generality. Differentiation of yields which in turn is equivalent to the convolution identity where . Conversely, if a DF satisfies (2) where is a measure having a finite LST, then is the DF of an infdiv law. See Theorem 4.10 in [1] and the reference there to Steutel’s original formulation of this result. Suppose now that the first moment and let denote the DF of the length-biased (or size-biased) version of . Clearly , and hence is the DF of a random variable, say. So if denotes a random variable having the DF and is infdiv, then (2) has the random variable formulation where denotes equality in law and the random variables on the right-hand side are independent. (Note that it is always assumed that random variables occurring on the right-hand side of in-law equalities are independent.) The equality (3) underlines the fact that length-biasing is an increasing operation with respect to the stochastic order; if . Conversely, if (3) holds for some positive defect random variable , then is infdiv. See [2] for more on this result. The primary question we address is that if (3) holds, then is also infdiv? A secondary question is whether is infdiv? If it is, then clearly is infdiv. The primary question can be extended to asking whether arbitrary order length-biasing of an infdiv law also is infdiv. Here, if , then we define the order- length-biased version of by , where .