We present recursions for the total number, Sn, of mutations in a sample of n individuals, when the underlying genealogical tree of the sample is modelled by a coalescent process with mutation rate r>0. The coalescent is allowed to have simultaneous multiple collisions of ancestral lineages, which corresponds to the existence of large families in the underlying population model. For the subclass of Λ-coalescent processes allowing for multiple collisions, such that the measure Λ(dx)/x is finite, we prove that Sn/(nr) converges in distribution to a limiting variable, S, characterized via an exponential integral of a certain subordinator. When the measure Λ(dx)/x2 is finite, the distribution of S coincides with the stationary distribution of an autoregressive process of order 1 and is uniquely determined via a stochastic fixed-point equation of the form with specific independent random coefficients A and B. Examples are presented in which explicit representations for (the density of) S are available. We conjecture that Sn/E(Sn)→1 in probability if the measure Λ(dx)/x is infinite.

In this paper, we present the distribution of the coalescence time of two DNA sequences (or genes) subject to symmetric migration between two islands, and conditional on the observed number of segregating sites in the sequences. The distribution for the segregating-site pattern is also obtained. Some surprising results emerge when both genes are initially on the same island. First, the post-data mean coalescence time is shown to be dependent on the migration parameter, as opposed to the pre-data mean. Second, both the post-data density and expectation for the coalescence time are shown to converge, in the weak-migration limit, to the corresponding panmictic results, as opposed to the pre-data situation where there is convergence in the density but not in the expectation. Finally, it is shown that there is convergence in the weak-migration limit in the distribution of the number of segregating sites but not in the expectation and variance. Numerical and graphical results for samples of size greater than two are also presented.