## Abstract

A long genomic segment inherited by a pair of individuals from a single, recent common ancestor is said to be identical-by-descent (IBD). Shared IBD segments have numerous applications in genetics, from demographic inference to phasing, imputation, pedigree reconstruction, and disease mapping. Here, we provide a theoretical analysis of IBD sharing under Markovian approximations of the coalescent with recombination. We describe a general framework for the IBD process along the chromosome under the Markovian models (SMC/SMC'), as well as introduce and justify a new model, which we term the renewal approximation, under which lengths of successive segments are independent. Then, considering the infinite-chromosome limit of the IBD process, we recover previous results (for SMC) and derive new results (for SMC') for the mean number of shared segments longer than a cutoff and the fraction of the chromosome found in such segments. We then use renewal theory to derive an expression (in Laplace space) for the distribution of the number of shared segments and demonstrate implications for demographic inference. We also compute (again, in Laplace space) the distribution of the fraction of the chromosome in shared segments, from which we obtain explicit expressions for the first two moments. Finally, we generalize all results to populations with a variable effective size.

Original language | American English |
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Pages (from-to) | 35-48 |

Number of pages | 14 |

Journal | Theoretical Population Biology |

Volume | 97 |

DOIs | |

State | Published - 1 Nov 2014 |

Externally published | Yes |

### Bibliographical note

Funding Information:We thank Asger Hobolth for commenting on the manuscript and for pointing out a number of arguments regarding Markov chains reversibility and detailed balance. S. C. thanks Eli Barkai, whose lecture notes on renewal theory have been heavily consulted, and the Human Frontier Science Program for financial support. I. P. thanks NIH grant 1R01MH095458-01A1 .

Publisher Copyright:

© 2014 Elsevier Inc.

## Keywords

- Coalescent theory
- IBD sharing
- Recombination
- Renewal theory
- SMC
- SMC'