Asexual Versus Promiscuous Bisexual Galton-Watson Processes: The Extinction Probability Ratio

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.