Self-Decomposable Laws from Continuous Branching Processes

The martingale limit law of the supercritical continuous time and state branching process either is compound Poisson or self-decomposable. This paper explores some general aspects of the latter case. A fundamental question for the latter case is whether the cumulant function of the martingale limit is a Thorin–Bernstein function. We make some progress by showing that it is special Bernstein if the cumulant function of the generating subordinator is special Bernstein. A specific parametric family of martingale limit cumulant functions is shown to be Thorin–Bernstein. Two complementary proofs of this fact are offered, one of which entirely avoids complex variable issues. The principal Lambert W-function is a boundary case of this family, thereby giving a new proof that it too is Thorin–Bernstein. Tail estimates of the distribution functions for this family are derived along with the right-hand tail and integral representations of their Lévy densities.