Computing the Density of the Kesten–Stigum Limit in Supercritical Galton–Watson Processes

Abstract

Abstract. This paper proposes a novel numerical method for computing the density of the Kesten–Stigum limit random variable associated with a supercritical Galton–Watson process. This random variable captures the cumulative effect of early demographic fluctuations and determines the random amplitude governing long-term exponential population growth. Beyond classical branching process theory, the Kesten–Stigum limit plays a central role in scaling limits of density-dependent population models, where it induces random initial conditions for deterministic fluid approximations, and in statistical inference problems for growing populations. Despite its importance, computing its density in a stable and efficient manner for general offspring distributions remains a significant challenge. Our approach leverages the functional equation satisfied by the Laplace–Stieltjes transform of the limit distribution and combines it with a moment-matching reconstruction based on Laguerre polynomials with exponential damping. The resulting method is computationally efficient and applies to offspring distributions with bounded support. Its effectiveness is demonstrated on several numerical examples, including biologically motivated case studies.