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Additional aspects of the generalized linear-fractional branching process

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Abstract

We derive some additional results on the Bienyamé–Galton–Watson-branching process with \(\theta \)-linear fractional branching mechanism, as studied by Sagitov and Lindo (Branching Processes and Their Applications. Lecture Notes in Statistics—Proceedings, 2016). This includes the explicit expression of the limit laws in both the subcritical cases and the supercritical cases with finite mean, and the long-run behavior of the population size in the critical case, limits laws in the supercritical cases with infinite mean when the \(\theta \) process is either regular or explosive, and results regarding the time to absorption, an expression of the probability law of the \(\theta \)-branching mechanism involving Bell polynomials, and the explicit computation of the stochastic transition matrix of the \(\theta \) process, together with its powers.

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Notes

  1. Throughout this work, a pgf will, therefore, be a function \(\phi \) which is absolutely monotone on (0, 1) with all non-negative derivatives of any order there, obeying \(\phi (1) \le 1.\)

  2. It is assumed here that \(i>1\). If \(i=1\), the condition on \(\rho -\phi ^{\circ n} (0)\) is no longer valid, but the one on \(\epsilon \) still is.

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Acknowledgments

T. Huillet acknowledges partial support from the “Chaire Modélisation mathématique et biodiversité”. N. Grosjean and T. Huillet also acknowledge support from the labex MME-DII Center of Excellence (Modèles mathématiques et économiques de la dynamique, de l’incertitude et des interactions, ANR-11-LABX-0023-01 project).

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  1. Laboratoire de Physique Théorique et Modélisation, CNRS-UMR 8089 et Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302, Cergy-Pontoise, France

    Nicolas Grosjean & Thierry Huillet

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  1. Nicolas Grosjean
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  2. Thierry Huillet
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Correspondence to Thierry Huillet.

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Grosjean, N., Huillet, T. Additional aspects of the generalized linear-fractional branching process. Ann Inst Stat Math 69, 1075–1097 (2017). https://doi.org/10.1007/s10463-016-0573-x

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  • DOI: https://doi.org/10.1007/s10463-016-0573-x

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