A Delayed Yule Process

Abstract:

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 continuous time Yule processes parameterized by $0 < \alpha \le 1$ and find a representation of the Poisson process as a delayed Yule process at delay rate $\alpha = {1/2}$. Moreover we extend Kendall’s limit theorem to include a larger class of positive martingales derived from functionals that gauge the population genealogy. 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. A connection with fixed points of the Holley-Liggett smoothing transformation also emerges in this context, about which much is known from general theory in terms of moments, tail decay, and so on.