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

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.