On the genealogical structure of critical branching processes in a varying environment

TL;DR

This paper investigates the genealogical structures of critical branching processes in varying environments, confirming their similarity to classical Galton-Watson processes, especially regarding the most recent common ancestor given survival.

Contribution

It extends the understanding of genealogical structures to varying environments, aligning them with well-known properties of Galton-Watson processes.

Findings

Genealogical structures are similar to those of Galton-Watson processes.

Analysis of the most recent common ancestor given survival.

Reduced branching processes behave consistently with classical models.

Abstract

Critical branching processes in a varying environment behave much the same as critical Galton-Watson processes. In this note we like to confirm this finding with regard to the underlying genealogical structures. In particular, we consider the most recent common ancestor given survival and the corresponding reduced branching processes, in the spirit of Zubkov (1975) and Fleischmann and Siegmund-Schultze (1977).