Computing the density of the Kesten-Stigum limit in supercritical Galton-Watson processes
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915988424884224 |
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| author | Cortinovis, Alice Hautphenne, Sophie Massei, Stefano |
| author_facet | Cortinovis, Alice Hautphenne, Sophie Massei, Stefano |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19633 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Computing the density of the Kesten-Stigum limit in supercritical Galton-Watson processes Cortinovis, Alice Hautphenne, Sophie Massei, Stefano Probability Numerical Analysis 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. |
| title | Computing the density of the Kesten-Stigum limit in supercritical Galton-Watson processes |
| topic | Probability Numerical Analysis |
| url | https://arxiv.org/abs/2601.19633 |