Almost sure asymptotics for the number of types for simple Ξ - coalescents

Let Kn be the number of types in the sample {1, . . . , n} of a Ξ-coalescent Π = (Πt)t≥0 with mutation and mutation rate r > 0. Let Π be the restriction of Π to the sample. It is shown that Mn/n, the fraction of external branches of Π (n) which are affected by at least one mutation, converges almost surely and in L (p ≥ 1) to M := ∫∞ 0 reStdt, where St is the fraction of singleton blocks of Πt. Since for coalescents without proper frequencies, the effects of mutations on non-external branches is neglectible for the asymptotics of Kn/n, it is shown that Kn/n → M for n → ∞ in L (p ≥ 1). For simple coalescents, this convergence is shown to hold almost surely. The almost sure results are based on a combination of the Kingman correspondence for random partitions and strong laws of large numbers for weighted i.i.d. or exchangeable random variables.