Approximate Modeling for Supercritical Galton-Watson Branching Processes with Compound Poisson-Gamma Distribution
Abstract
We study asymptotic properties of supercritical Galton-Watson (GW) branching processes in the asymptotic where the mean of the offspring distribution approaches 1 from above. We show that the population-size distribution of the GW branching processes at a sufficiently large generation in this asymptotic can be approximated by a compound Poisson-gamma distribution. Numerical experiments revealed that the compound Poisson-gamma models were in good agreement with the corresponding GW models for sufficiently large generations under a reasonable parameter regime. Our results can be regarded as supporting the use of the compound Poisson-gamma model as a model for cascaded multiplication processes, such as detection signals of electron multipliers and population sizes of individuals with specific biological characteristics.
Keywords
Cite
@article{arxiv.2509.15502,
title = {Approximate Modeling for Supercritical Galton-Watson Branching Processes with Compound Poisson-Gamma Distribution},
author = {Kyoya Uemura and Tomoyuki Obuch and Toshiyuki Tanaka},
journal= {arXiv preprint arXiv:2509.15502},
year = {2026}
}
Comments
18 pages, 5 figures, to be submitted to Physical Review X. Supplementary material is available in the file "Supplemental_Material.pdf"