Connecting Yule Process, Bisection and Binary Search Tree via Martingales

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 $0 < α\le 1$ that includes the Poisson process at $α= {1/2}$. 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 surpr…