Asymptotic line-of-descent distributions

Asymptotic Distributions of Coalescence Times and Ancestral Lineage Numbers for Populations with Temporally Varying Size

“…For a sample of size n, denote by T m the mth coalescent time, when m + 1 lineages coalesce into m lineages, and A n (t) the number of ancestral lineages at time t back from the current generation. Similar to the results in Griffiths (1984), the number of ancestral lineages, A n (t), and the coalescence times, T m , are asymptotically normal, with the mean and variance of these distributions depending on the population size function, N(t). At the very early stage of the coalescent, when t / 0, the number of coalesced lineages n 2 A n (t) follows a Poisson distribution, and as m / n, nðn 2 1ÞT m =2Nð0Þ follows a gamma distribution.…”

“…Denote by A n (t) the number of ancestral lineages at t generations back in the past. In an ancestral process where both coalescent and mutation events can reduce the number of lineages, A n (t) is referred to as the number of nonmutant lineages at t generations back from the present (Griffiths 1984). In this context, we consider the genealogical history of only coalescent events, in which mutations are treated separately and assumed to occur independently following a Poisson process along the branches of a given gene genealogy.…”

“…Both exact distributions of coalescence times and ancestral lineage numbers are expressed as the sum of alternating series, and the terms in the series become numerically intractable for large samples. More computationally attractive are their asymptotic distributions, which were derived in Griffiths (1984) for populations with constant size. In this article, we derive the asymptotic distributions of coalescence times and ancestral lineage numbers for populations with temporally varying size.…”

“…The asymptotic theories of the coalescence times and the number of ancestral lineages for large gene genealogies in constant populations have been derived by Griffiths (1984). He demonstrated that as t / 0 and the sample size n / N, the distributions of A n (t) and T m converge asymptotically to normal distributions.…”

“…However, if we scale the time to account for the fluctuation in population size by R t 0 ðds=NðsÞÞ; t $ 0, where N(Á) is the population size function over time, the coalescent process on the new time scale is equivalent to the standard coalescent (Kingman 1982b;Griffiths and Tavaré 1994b). The theorems for the standard coalescent in Griffiths (1984) can then be borrowed to obtain asymptotic distributions for populations with temporally varying size. Extension of Griffiths' theorems to populations with time-varying size is very important for population genetic inference, since most ancestral inference is based on the nonequilibrium genetic polymorphism patterns in populations with temporally varying size.…”