On a coalescence process and its branching genealogy

TL;DR

This paper introduces a coalescent process modeled as a recursive box-filling mechanism with a genealogy described by a time-inhomogeneous Bienyame-Galton-Watson process, analyzing its expected box size and emptiness probability.

Contribution

It provides a novel recursive coalescent model linked to a specific genealogical process and derives exact asymptotics for special cases with linear-fractional or quadratic mechanisms.

Findings

Derived asymptotic behaviors for the coalescent process.

Analyzed expected size and emptiness probability of boxes.

Explored special cases with exact computations.

Abstract

We define and analyze a coalescent process as a recursive box-filling process whose genealogy is given by an ancestral time-reversed, time-inhomogeneous Bienyam\'{e}-Galton-Watson process. Special interest is on the expected size of a typical box and its probability of being empty. Special cases leading to exact asymptotic computations are investigated when the coalescing mechanisms are either linear-fractional or quadratic.