Geometric branching process: time to extinction

Fisher, Evan; Zhou, Zhiqi

doi:10.2140/involve.2025.18.601

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Geometric branching process: time to extinction

Evan Fisher and Zhiqi Zhou

Vol. 18 (2025), No. 4, 601–612

DOI: 10.2140/involve.2025.18.601

Abstract

We derive a closed form expression for the expected time to extinction for the Galton–Watson–Bienaymé process with geometric offspring distribution in terms of the $q$ -digamma function. This forms the basis for simply expressed tight upper and lower bounds and an asymptotic result as the mean of the offspring distribution approaches $1$ . We also obtain a result on the variance of the time to extinction.

Keywords

geometric branching process, extinction time, digamma function, geometric distribution

Mathematical Subject Classification

Primary: 60J80

Supplementary material

Appendices with Mathematica code and output for statistics of the time to extinction

Milestones

Received: 14 March 2023

Revised: 19 January 2024

Accepted: 25 January 2024

Published: 30 July 2025

Communicated by Anant Godbole

Authors

Evan Fisher
Department of Mathematics Lafayette College Easton, PA United States

Zhiqi Zhou
Department of Mathematics Lafayette College Easton, PA United States	Chicago, IL United States

Vol. 18, No. 4, 2025