Kin-number distributions over age, sex, and time

Mathematical kinship demography is an expanding area of research. Most models explore the expected number of kin without accounting for demographic stochasticity. Recently, a paper provided a method to calculate the complete number-distribution of kin in a one-sex time-invariant demography. We extend this method to the case of two-sexes and to time-variant demographic rates. Drawing from the mathematical tools of Fourier and convolution theory as well as basic probability and matrix algebra, we derive closed form expressions which capture the recursive nature of kin replen-ishment, generation-by-generation. Formulae presented here extend arbitrary genealogical distances to recover relatives considered in the leading frameworks of kinship. All we require as inputs are age, sex, and time-specific mortality and fertility schedules. This research presents the first kinship model able to predict the probable numbers of relatives, structured by age and sex within a time-varying demography. As well as producing the probable numbers of living kin, the model flexibly extends to give the probable numbers of deaths an individual experiences. Such a detailed analysis of the kin-network will be useful in many fields.