A Delayed Yule Process

In now classic work, David Kendall (1966) recognized that the Yule process and Poisson process could be related by a (random) time change. Furthermore, he showed that the Yule population size rescaled by its mean has an almost sure exponentially distributed limit as $t\to \infty$. In this note we introduce a class of coupled delayed Yule processes parameterized by $0 < \alpha \le 1$ that includes the Poisson process at $\alpha = {1/2}$. Moreover we extend Kendall's limit theorem to include a larger class of positive martingales derived from functionals that gauge the population genealogy. A somewhat surprising connection with the Holley-Liggett smoothing transformation also emerges in this context. Specifically, the latter is exploited to uniquely characterize the moment generating functions of distributions of the limit martingales, generalizing Kendall's mean one exponential limit.