extinction minus one

The riddle from The Riddler of 19 Feb. is about the Bernoulli Galton-Watson process, where each individual in the population has one or zero descendant with equal probabilities: Starting with a large population os size N, what is the probability that the size of the population on the brink of extinction is equal to one? While it is easy to show that the probability the n-th generation is extinct is

\mathbb{P}(S_n=0) = 1 - \frac{1}{2^{nN}}

I could not find a way to express the probability to hit one and resorted to brute force simulation, easily coded

for(t in 1:(T<-1e8)){N=Z=1e4 
while(Z>1)Z=rbinom(1,Z,.5)
F=F+Z}
F/T

which produces an approximate probability of 0.7213 or 0.714. The impact of N is quickly vanishing, as expected when the probability to reach 1 in one generation is negligible…

However, when returning to Dauphine after a two-week absence, I presented the problem with my probabilist neighbour François Simenhaus, who immediately pointed out that this probability was more simply seen as the probability that the maximum of N independent geometric rv’s was achieved by a single one among the N. Searching later a reference for that probability, I came across the 1990 paper of Bruss and O’Cinneide, which shows that the probability of uniqueness of the maximum does not converge as N goes to infinity, but rather fluctuates around 0.72135 with logarithmic periodicity. It is only when N=2^n that the sequence converges to 0.721521… This probability actually writes down in closed form as

N\sum_{i=1}^\infty 2^{-i-1}(1-2^{-i})^{N-1}

(which is obvious in retrospect!, albeit containing a typo in the original paper which is missing a ½ factor in equation (17)) and its asymptotic behaviour is not obvious either, as noted by the authors.

On the historical side, and in accordance with Stiegler’s law, the Galton-Watson process should have been called the Bienaymé process! (Bienaymé was a student of Laplace, who successively lost positions for his political idea, before eventually joining Académie des Sciences, and later founding the Société Mathématique de France.)

2 Responses to “extinction minus one”

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