A renewal theory approach to IBD sharing
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(2014)cite arxiv:1403.1325.

Long genomic segments that are nearly identical between a pair of individuals and are inherited from a recent common ancestor without recombination are called identical-by-descent (IBD) segments. IBD sharing has 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 average fraction of the chromosome found in long shared segments and the average number of such segments. A number of new results for tree heights in SMC' are proved as lemmas. 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 use renewal theory to compute the distribution of the fraction of the chromosome shared. While the expression is again in Laplace space, we could invert the first two moments and compare a number of approximations. Finally, we generalized all results to populations with variable historical effective size.
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