On the number of allelic types for samples taken from exchangeable coalescents with mutation

Abstract

Let Kn denote the number of types of a sample of size n taken from an exchangeable coalescent process (Ξ-coalescent) with mutation. A distributional recursion for the sequence (Kn) n∈N is derived. If the coalescent does not have proper frequencies, i.e., if the characterizing measure Ξ on the infinite simplex ∆ does not have mass at zero and satisfies R

Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson–Dirichlet coalescent

“…The method of moments yields the convergence M exter nal n /n → r S in distribution as n → ∞. The convergence M inter nal n /n → 0 in L 1 follows (in analogy to the proof of Theorem 3) from the fact (see [12,Lemma 4.1]) that the number of collisions C n satisfies C n /n → 0 in L 1 as n → ∞.…”

“…For n ∈ N consider the restricted coalescent process Π (n) = (Π (n) t ) t≥0 := ( n • Π t ) t≥0 . From the paintbox construction of the coalescent it follows (see, for example, [12]) that its block…”

“…There is (see, for example, [12]) the following interpretation of the external branch lengths τ i , i ∈ N, in terms of the frequencies of singletons S t , t ≥ 0. Let t 1 , .…”

“…The stated uniqueness follows by differentiating the functional equation (12) k times with respect to λ and taking the limit as λ → 0.…”

Linearization of the Kingman Coalescent

“…Both approaches model proliferation of lineages over time. Further examples include β-coalescent [32], Λ-coalescent [33,34], Ξ-coalescent [35,36], and Galton-Watson theory [37,38]. Technical mathematical treatments tend to assume the foundations of ancestral processes.…”