Asymptotics of continuous-time discrete state space branching processes for large initial state

TL;DR

This paper investigates the asymptotic behavior of continuous-time discrete-state branching processes as the initial population size grows large, revealing different limit processes based on offspring distribution moments.

Contribution

It provides a comprehensive analysis of the scaling limits and transfer results for offspring distribution generating functions in large initial state regimes.

Findings

Limits are Gaussian, Ornstein-Uhlenbeck, or continuous-state branching processes depending on offspring moments.

Transfer results connect offspring generating functions to process distributions.

Asymptotic relations for generating functions carry over to process distributions.

Abstract

Scaling limits for continuous-time branching processes with discrete state space are provided as the initial state tends to infinity. Depending on the finiteness or non-finiteness of the mean and/or the variance of the offspring distribution, the limits are in general time-inhomogeneous Gaussian processes, time-inhomogeneous generalized Ornstein-Uhlenbeck type processes or continuous-state branching processes. We also provide transfer results showing how specific asymptotic relations for the probability generating function of the offspring distribution carry over to those of the one-dimensional distributions of the branching process.