Self-Decomposability and the Lambert Law

The Lambert W is a Bernstein function and hence the cumulant function of a positive infinitely divisible law, the Lambert law L(Λ). The Lambert law is self-decomposable with a unit rate compound Poisson background driving Lévy process (BDLP) with a certain jump law L(J). Thus, it satisfies the random perpetuity relation Λ=LU(Λ+J), where U has the standard uniform law, as well as the relation Λ=LUSJ, where S is the stationary-excess operator. If the law of J is unspecified, then these two in-law relations together characterise the law of Λ. Generalisations are pursued in which the Poisson jump rate a is arbitrary. If a≠1, the above two conditions determine a family of positive infinitely divisible laws for which the Laplace–Stieltjes transforms satisfy an algebraic trinomial equation investigated by Lambert and Euler. These laws occur as limiting distributions of shot noise processes and continuous-time branching processes. A sequence of n-times self-decomposable positive laws derived from a given subordinator is called a BDLP ladder. The BDLP ladder that includes the Lambert law has cumulant functions which are polynomial forms of the Lambert function. Under moment conditions, general BDLP ladder laws are asymptotically normal or stable.