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Geometric branching process: time to extinction

Evan Fisher and Zhiqi Zhou

Vol. 18 (2025), No. 4, 601–612
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