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arXiv · 2601.19633

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

Alice Cortinovis·Sophie Hautphenne·Stefano Massei
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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.

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Alice Cortinovis, Sophie Hautphenne, Stefano Massei. 2026-05-07. Computing the density of the Kesten-Stigum limit in supercritical Galton-Watson processes. https://arxiv.org/abs/2601.19633

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