Limiting distribution of X-chromosomal coalescence times under first-cousin consanguineous mating

By providing additional opportunities for coalescence within families, the presence of consanguineous unions in a population reduces coalescence times relative to non-consanguineous populations. First-cousin consanguinity can take one of six forms differing in the configuration of sexes in the pedigree of the male and female cousins who join in a consanguineous union: patrilateral parallel, patrilateral cross, matrilateral parallel, matrilateral cross, bilateral parallel, and bilateral cross. Considering populations with each of the six types of first-cousin consanguinity individually and a population with a mixture of the four unilateral types, we examine coalescent models of consanguinity. We previously computed, for first-cousin consanguinity models, the mean coalescence time for X-chromosomal loci and the limiting distribution of coalescence times for autosomal loci. Here, we use the separation-of-time-scales approach to obtain the limiting distribution of coalescence times for X-chromosomal loci. This limiting distribution has an instantaneous coalescence probability that depends on the probability that a union is consanguineous; lineages that do not coalesce instantaneously coalesce according to an exponential distribution. We study the effects on the coalescence time distribution of the type of first-cousin consanguinity, showing that patrilateral-parallel and patrilateral-cross consanguinity have no effect on X-chromosomal coalescence time distributions and that matrilateral-parallel consanguinity decreases coalescence times to a greater extent than does matrilateral-cross consanguinity.