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The genealogy of branching processes and the age of our most recent common ancestor. (English) Zbl 0837.60080

The first part of the paper discusses limiting results for a sequence of nearly-critical continuous-time Markov branching processes. In all processes the mean life is one and the offspring variance is \(\sigma^2\), but in the \(t\)th process the mean is \(1+ \alpha/t\). The limiting behaviour as \(t\) goes to infinity is considered. First, theorems are established on the extinction probability and the exponential limit law. Next, the reduced process is considered; this is formed by looking at the process formed, as \(s\) varies, by the number of individuals at time \(s\) having descendants alive at time \(t\) (the same \(t\) as labels the sequence!). It is shown to converge to a pure birth process. This result is then used to give information on the time of death of the most recent common ancestor of two randomly chosen individuals at time \(t\). The second part of the paper looks at the application of the results to derive an estimate of the time until the first common ancestor of all women, based on data from mitochondrial DNA. This part provides a delightful complement to the elegant theoretical manoeuvres that preceded it.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J85 Applications of branching processes
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