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

Baake, Ellen; Lenz, Ute; Wakolbinger, Anton

Electron. Commun. Probab. 21, Paper No. 59, 16 p. (2016).

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.

MSC:

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

Keywords:

common ancestor type distribution; ancestral selection graph; lookdown graph; pruning; \(\Lambda\)-Wright-Fisher diffusion; selection; mutation; strong pathwise siegmund duality; flights

Citations:

Zbl 1342.92141