Abstract
It is well known that a simple, supercritical Bienaymé-Galton-Watson process turns into a subcritical such process, if conditioned to die out. We prove that the corresponding holds true for general, multi-type branching, where child-bearing may occur at different ages, life span may depend upon reproduction, and the whole course of events is thus affected by conditioning upon extinction.
Highlights
The theory of branching processes was born out of Galton’s famous family extinction problem
Extinction has retaken a place in the foreground, for reasons from both conservation and evolutionary biology
The question arises whether general branching populations bound for extinction must behave like subcritical populations
Summary
The theory of branching processes was born out of Galton’s famous family extinction problem. The question arises whether (non-critical) general branching populations ( known as CrumpMode-Jagers, or CMJ, processes) bound for extinction must behave like subcritical populations We answer this in the affirmative: a general, multi-type branching process conditioned to die out, remains a branching process, but one almost surely dying out. The branching property can be summarised into the fact that given her type and birth time, the daughter process of any individual born is independent of all individuals not in her progeny (into which she herself is included). In all other respects her life is independent of all others, once her type is given This reestablishes the branching character, but with a suitably amended probability measure over her life career, which clearly is non-supercritical in the sense that the probability of ultimate extinction is one, from any starting type that can be realised
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
