We consider the bisexual Galton-Watson process (BGWP) with
promiscuous mating, that is, a branching process which behaves like an ordinary
Galton-Watson process as long as at least one male is produced in each
generation. For the case of Poissonian reproduction, it was pointed out by
Daley, Hull and Taylor 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) providing the number of
ancestors grows to infinity. They further mentioned that they had no
theoretical justification for this phenomenon. In the present article we will
prove upper and lower bounds for the ratio between the two extinction
probabilities and introduce a recursive algorithm that can easily be
implemented on a computer to produce very accurate approximations for that
ratio. The final section contains a number of numerical results that have been
obtained by use of this algorithm.
Publié le : 1996-08-14
Classification:
Bisexual Galton-Watson process,
promiscuous mating,
extinction probability,
killed Markov process,
harmonic function,
function iteration,
60J80
@article{1034968234,
author = {Alsmeyer, Gerold and R\"osler, Uwe},
title = {The bisexual Galton-Watson process with promiscuous mating:
extinction probabilities in the supercritical case},
journal = {Ann. Appl. Probab.},
volume = {6},
number = {1},
year = {1996},
pages = { 922-939},
language = {en},
url = {http://dml.mathdoc.fr/item/1034968234}
}
Alsmeyer, Gerold; Rösler, Uwe. The bisexual Galton-Watson process with promiscuous mating:
extinction probabilities in the supercritical case. Ann. Appl. Probab., Tome 6 (1996) no. 1, pp. 922-939. http://gdmltest.u-ga.fr/item/1034968234/