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Bruss, How to apply a medicament, if its direct efficiency is unknown, Ann. Bruss, Branching processes with random absorbing processes, J. online, 2021, preprint version available at, 2021. Curbelo, Weak independence of events and the converse of the Borel-Cantelli lemma, Expo. Zucca, Local and global survival for nonhomogeneous random walk systems on $\mathbb Z$, Adv. Barndorff-Nielsen, On Partial, Extrema Independent, Identically Distributed Random Variables and Amanuensis Mathematics, Techn. Vatutin, Criticality for branching processes in random environment, Ann. Hautphenne, A pathwise approach to the extinction of branching processes with countably many types, Stoch. Finally, we briefly discuss relatively complicated resource-dependent branching processes to show that, again, using Galton-Watson reproduction schemes (whenever reasonable) can be a convincing approach to new processes, which are then sufficiently tractable to obtain results of interest.ġ. To gain more generality we then look at bisexual Galton-Watson processes. Further questions bring us, via the Borel-Cantelli lemma, to $\varphi$-branching processes and extensions. Then we pass to random absorbing processes, and also recall and discuss a problem in medicine. We start with a discussion on a controlled Galton-Watson process. The thread of the article is the role which the Galton-Watson process had played in the author's own research. It is at the same time the author's way of honoring two distinguished scientists in this domain, both from the Russian Academy of Sciences, and congratulating them on their special birthdays.
But the probability of survival of a new type may be quite low even if λ > 1 and the population as a whole is experiencing quite strong exponential increase. For λ ≤ 1 eventual extinction will occur with probability 1. Galton-Watson survival probabilities for different exponential rates of population growth, if the number of children of each parent node can be assumed to follow a Poisson distribution.