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February 2002 Asexual Versus Promiscuous Bisexual Galton-Watson Processes: The Extinction Probability Ratio
Gerold Alsmeyer, Uwe Rösler
Ann. Appl. Probab. 12(1): 125-142 (February 2002). DOI: 10.1214/aoap/1015961158

Abstract

We consider the supercritical bisexual Galton–Watson process (BGWP) with promiscuous mating, that is, a branching process which behaves like an ordinary supercritical Galton–Watson process (GWP) as long as at least one male is born in each generation. For a certain example, it was pointed out by Daley, Hull and Taylor [J. Appl. Probab. 23 (1986) 585–600] that the extinction probability of such a BGWP apparently behaves like a constant times the respective probability of its asexual counterpart (where males do not matter) if the number of ancestors grows to $\infty$. In an earlier paper, we provided general upper and lower bounds for the ratio between both extinction probabilities and also numerical results that seemed to confirm the convergence of that ratio. However, theoretical considerations rather led us to the conjecture that this does not generally hold. The present article turns this conjecture into a rigorous result. The key step in our analysis is to identify the extinction probability ratio as a certain functional of a subcritical ordinary GWP and to prove its continuity as a function of the number of ancestors in a suitable topology associated with the entrance Martin boundary of that GWP.

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Gerold Alsmeyer. Uwe Rösler. "Asexual Versus Promiscuous Bisexual Galton-Watson Processes: The Extinction Probability Ratio." Ann. Appl. Probab. 12 (1) 125 - 142, February 2002. https://doi.org/10.1214/aoap/1015961158

Information

Published: February 2002
First available in Project Euclid: 12 March 2002

zbMATH: 1020.60073
MathSciNet: MR1890059
Digital Object Identifier: 10.1214/aoap/1015961158

Subjects:
Primary: 60J80
Secondary: 60J45

Keywords: Bisexual Galton-Watson process , extinction probability , promiscuous mating , quasi-invariant measure , Time reversal

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.12 • No. 1 • February 2002
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