A scaling limit theorem for Galton-Watson processes in varying environments

TL;DR

This paper establishes a scaling limit theorem for Galton-Watson processes in changing environments, linking discrete models to continuous-state branching processes under specific conditions.

Contribution

It provides a simple sufficient condition for weak convergence of these processes in the Skorokhod space, advancing understanding of their limiting behavior.

Findings

Weak convergence criterion in terms of probability generating functions

Connection between discrete Galton-Watson processes and continuous-state branching processes

Extension of previous results to varying environments

Abstract

We prove a scaling limit theorem for discrete Galton-Watson processes in varying environments. A simple sufficient condition for the weak convergence in the Skorokhod space is given in terms of probability generating functions. The limit theorem gives rise to the continuous-state branching processes in varying environments studied recently by several authors.