Distribution of genome shared IBD by half-sibs: approximation by the Poisson clumping heuristic

The distribution of the proportion of genome shared identical by descent (IBD) by c half-sibs is calculated for a general c. With data availability the consideration of the chromosome as a whole rather than discrete loci becomes of increasing practical importance. Assuming the genomic continuum model, which allows for recombination, the crossover process in a chromosome pedigree is viewed as a continuous-time Markov random walk on the vertices of a hypercube with time parameter map distance along the chromosome. The desired distribution corresponds to the distribution of sojourn times of the process in a small set of vertices. The sojourn times are well approximated via the Poisson clumping heuristic. It is shown explicitly that there is no "equivalent" number of independently segregating loci that will yield the same results as with the genomic continuum model. Results are given for the human genome.