Total length of the genealogical tree for quadratic stationary continuous-state branching processes

TL;DR

This paper establishes the existence and properties of the total length process of genealogical trees in quadratic stationary continuous-state branching processes, providing new insights into their structure and fluctuations.

Contribution

It introduces the total length process for genealogical trees in quadratic stationary CSBPs and analyzes its Laplace transform and fluctuation properties, extending prior work on constant size populations.

Findings

Derived the Laplace transform of the total length process.

Proved the fluctuation behavior of the total length.

Described the lineage tree structure and time reversal properties.

Abstract

We prove the existence of the total length process for the genealogical tree of a population model with random size given by a quadratic stationary continuous-state branching processes. We also give, for the one-dimensional marginal, its Laplace transform as well as the fluctuation of the corresponding convergence. This result is to be compared with the one obtained by Pfaffelhuber and Wakolbinger for constant size population associated to the Kingma coalescent. We also give a time reversal property of the number of ancestors process at all time, and give a description of the so-called lineage tree in this model.