Scaling limits of bisexual Galton-Watson processes

TL;DR

This paper investigates the scaling limits of bisexual Galton-Watson processes, establishing tightness and identifying possible stochastic differential equation limits, with applications to mutual fidelity scenarios.

Contribution

It introduces a rigorous framework for analyzing the asymptotic behavior of bisexual Galton-Watson processes and characterizes their limits via stochastic systems.

Findings

Proves tightness of rescaled processes

Identifies limit as solutions to stochastic systems

Demonstrates convergence under certain conditions

Abstract

Bisexual Galton-Watson processes are discrete Markov chains where reproduction events are due to mating of males and females. Owing to this interaction, the standard branching property of Galton-Watson processes is lost. We prove tightness for conveniently rescaled bisexual Galton-Watson processes, based on recent techniques developed by Bansaye, Caballero and M{\'e}l{\'e}ard. We also identify the possible limits of these rescaled processes as solutions of a stochastic system, coupling two equations through singular coefficients in Poisson terms added to square roots as coefficients of Brownian motions. Under some additional integrability assumptions, pathwise uniqueness of this limiting system of stochastic differential equations and convergence of the rescaled processes are obtained. Two examples corresponding to mutual fidelity are considered.