The Feller diffusion as the limit of a coalescent point process

TL;DR

This paper investigates the Feller diffusion as a limit of a coalescent point process with a skewed node height distribution, unifying recent results on branching process scaling limits and extending sampling concepts.

Contribution

It introduces a unified approach to interpret scaling limits of branching processes as properties of the Feller diffusion and extends Bernoulli sampling to the diffusion limit for finite Poisson samples.

Findings

Coalescent tree of Poisson-sampled Feller diffusion matches a coalescent point process.

Methods for analyzing k-sampled birth-death processes are adapted for Feller diffusion.

The algebraic form of node height distribution is consistent across sampling methods.

Abstract

The Feller diffusion is studied as the limit of a coalescent point process in which the density of the node height distribution is skewed towards zero. Using a unified approach, a number of recent results pertaining to scaling limits of branching processes are reinterpreted as properties of the Feller diffusion arising from this limit. The notion of Bernoulli sampling of a finite population is extended to the diffusion limit to cover finite Poisson-distributed samples drawn from infinite continuum populations. We show that the coalescent tree of a Poisson-sampled Feller diffusion corresponds to a coalescent point process with a node height distribution taking the same algebraic form as that of a Bernoulli-sampled birth-death process. By adapting methods for analysing k-sampled birth-death processes, in which the sample size is pre-specified, we develop methods for studying the coalescent properties of the k-sampled Feller diffusion.