Approximation of the number of descendants in branching processes

TL;DR

This paper presents simple approximation methods for the relative limit densities of descendants in Galton--Watson processes, leveraging Fourier coefficients and binomial coefficients, with demonstrated accuracy through numerical examples.

Contribution

It introduces novel approximation techniques based on Fourier and binomial coefficients for descendant densities in branching processes.

Findings

Approximations closely match exact values in numerical tests.

Fourier coefficient decay enables effective approximation.

Method simplifies analysis of descendant distributions.

Abstract

We discuss approximations of the relative limit densities of descendants in Galton--Watson processes that follow from the Karlin--McGregor near-constancy phenomena. These approximations are based on the fast exponentially decaying Fourier coefficients of Karlin--McGregor functions and the binomial coefficients. The approximations are sufficiently simple and show good agreement between approximate and exact values, which is demonstrated by several numerical examples.