The common ancestor type distribution of a \(\Lambda\)-Wright-Fisher process with selection and mutation. (English) Zbl 1346.60131

Summary: Using graphical methods based on a ‘lookdown’ and pruned version of the ancestral selection graph, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population. This extends results from [U. Lenz et al., Theor. Popul. Biol. 103, 27–37 (2015; Zbl 1342.92141)] to the case of heavy-tailed offspring, directed by a reproduction measure \(\Lambda\). The representation is in terms of the equilibrium tail probabilities of the line-counting process \(L\) of the graph. We identify a strong pathwise Siegmund dual of \(L\), and characterise the equilibrium tail probabilities of \(L\) in terms of hitting probabilities of the dual process.


60J75 Jump processes (MSC2010)
92D15 Problems related to evolution
60C05 Combinatorial probability
05C80 Random graphs (graph-theoretic aspects)


Zbl 1342.92141
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