Computing the density of the Kesten-Stigum limit in supercritical Galton-Watson processes

TL;DR

This paper introduces a new numerical method to accurately compute the density of the limit random variable in supercritical Galton-Watson processes, addressing a key challenge in understanding long-term population growth.

Contribution

It develops a stable, efficient approach using a functional equation and moment-matching with Laguerre polynomials to approximate the density for arbitrary offspring laws.

Findings

Method accurately computes the density in several polynomial offspring cases.

Approach leverages the Laplace-Stieltjes transform and moment-matching techniques.

Validated on examples with polynomial offspring generating functions.

Abstract

This paper proposes a novel numerical method for computing the density of the limit random variable associated with a supercritical Galton-Watson process. This random variable captures the effect of early demographic fluctuations and determines the random amplitude of long-term exponential population growth. While the existence of a non-trivial limit is ensured by the Kesten-Stigum theorem, computing its density in a stable and efficient manner for arbitrary offspring laws remains a significant challenge. The proposed approach leverages a functional equation that characterizes the Laplace-Stieltjes transform of the limit distribution and combines it with a moment-matching method to obtain accurate approximations within a class of linear combinations of Laguerre polynomials with exponential damping. The effectiveness of the approach is validated on several examples in which the offspring generating function is a polynomial of bounded degree.