## Electronic Communications in Probability

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

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

Ellen Baake, Ute Lenz, and Anton Wakolbinger

#### 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**

Received: 11 March 2016

Accepted: 8 August 2016

First available in Project Euclid: 9 September 2016

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/16-ECP16

**Mathematical Reviews number (MathSciNet)**

MR3548771

**Zentralblatt MATH identifier**

1346.60131

**Subjects**

Primary: 60J75: Jump processes 92D15: Problems related to evolution 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20]

**Keywords**

common ancestor type distribution ancestral selection graph lookdown graph pruning $\Lambda$-Wright-Fisher diffusion selection mutation strong pathwise Siegmund duality flights

**Rights**

Creative Commons Attribution 4.0 International License.

#### 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