## The Annals of Statistics

- Ann. Statist.
- Volume 42, Number 4 (2014), 1361-1393.

### Markov jump processes in modeling coalescent with recombination

Xian Chen, Zhi-Ming Ma, and Ying Wang

#### Abstract

Genetic recombination is one of the most important mechanisms that can generate and maintain diversity, and recombination information plays an important role in population genetic studies. However, the phenomenon of recombination is extremely complex, and hence simulation methods are indispensable in the statistical inference of recombination. So far there are mainly two classes of simulation models practically in wide use: back-in-time models and spatially moving models. However, the statistical properties shared by the two classes of simulation models have not yet been theoretically studied. Based on our joint research with CAS-MPG Partner Institute for Computational Biology and with Beijing Jiaotong University, in this paper we provide for the first time a rigorous argument that the statistical properties of the two classes of simulation models are identical. That is, they share the same probability distribution on the space of ancestral recombination graphs (ARGs). As a consequence, our study provides a unified interpretation for the algorithms of simulating coalescent with recombination, and will facilitate the study of statistical inference on recombination.

#### Article information

**Source**

Ann. Statist., Volume 42, Number 4 (2014), 1361-1393.

**Dates**

First available in Project Euclid: 25 June 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1403715204

**Digital Object Identifier**

doi:10.1214/14-AOS1227

**Mathematical Reviews number (MathSciNet)**

MR3226160

**Zentralblatt MATH identifier**

1319.60163

**Subjects**

Primary: 60J25: Continuous-time Markov processes on general state spaces 65C60: Computational problems in statistics

Secondary: 92B15: General biostatistics [See also 62P10] 92D25: Population dynamics (general) 60J75: Jump processes

**Keywords**

Markov jump process coalescent process random sequence conditional distribution genetic recombination ancestral recombination graph back-in-time algorithm spatial algorithm

#### Citation

Chen, Xian; Ma, Zhi-Ming; Wang, Ying. Markov jump processes in modeling coalescent with recombination. Ann. Statist. 42 (2014), no. 4, 1361--1393. doi:10.1214/14-AOS1227. https://projecteuclid.org/euclid.aos/1403715204

#### Supplemental materials

- Supplementary material: Supplement to “Markov jump processes in modeling coalescent with recombination”. The supplementary file is divided into two Appendixes. Appendix A contains the proofs of Propositions 1–9 and Propositions 11–13. Appendix B is devoted to the calculation of the conditional distribution $P(T_{j+1}^{i+1}\in B,\xi_{j+1}^{i+1}=\vec{\xi}|X^{S_{i}},S_{i+1},T_{0}^{i+1},\xi^{i+1},\ldots,T_{j}^{i+1},\xi_{j}^{i+1})$. In particular, the proofs of Theorems 5, 6 and 7 are presented, respectively, in the proofs of Theorems B.10, B.11 and B.12 in Appendix B.Digital Object Identifier: doi:10.1214/14-AOS1227SUPP