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An ancestral recombination graph. (English) Zbl 0893.92020
Donnelly, Peter (ed.) et al., Progress in population genetics and human evolution. Proceedings of the workshop on Mathematical population genetics, held at the IMA, Minnesota, MN, USA, January 24–28, 1994. New York, NY: Springer. IMA Vol. Math. Appl. 87, 257-270 (1997).
Summary: This paper describes a model of a gene as a continuous length of DNA represented by the interval \([0,1]\). The ancestry of a sample of genes is complicated by possible recombination events, where a gene can have two parent genes. An analogue of Kingman’s coalescent process [J. F. C. Kingman, Stochastic Processes Appl. 13, 235-248 (1982; Zbl 0491.60076)], in which the ancestry of a sample of genes at a single locus is described by a stochastic binary tree, is a stochastic ancestral recombination graph, with vertices where coalescent or recombination events occur. All the information about ancestry is contained in this graph.
The sample DNA lengths have marginal ancestral trees at each point in \([0,1]\) which are imbedded in the graph. An upper bound is found for the expected number of distinct most recent common ancestors of these trees, and the expected maximum waiting time to these ancestors.
For the entire collection see [Zbl 0861.00030].

92D15 Problems related to evolution
60G35 Signal detection and filtering (aspects of stochastic processes)
05C90 Applications of graph theory