## Electronic Communications in Probability

### The common ancestor type distribution of a $\Lambda$-Wright-Fisher process with selection and mutation

#### Abstract

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 [17] 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.

#### Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 59, 16 pp.

Dates
Accepted: 8 August 2016
First available in Project Euclid: 9 September 2016

https://projecteuclid.org/euclid.ecp/1473424720

Digital Object Identifier
doi:10.1214/16-ECP16

Mathematical Reviews number (MathSciNet)
MR3548771

Zentralblatt MATH identifier
1346.60131

#### Citation

Baake, Ellen; Lenz, Ute; Wakolbinger, Anton. The common ancestor type distribution of a $\Lambda$-Wright-Fisher process with selection and mutation. Electron. Commun. Probab. 21 (2016), paper no. 59, 16 pp. doi:10.1214/16-ECP16. https://projecteuclid.org/euclid.ecp/1473424720

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