Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models

Abstract

Haploid population models with non-overlapping generations and fixed population size N are considered. It is assumed that the family sizes ν1,…,νN within a generation are exchangeable random variables. Rates of convergence for the finite-dimensional distributions of a properly time-scaled ancestral coalescent process are established and expressed in terms of the transition probabilities of the ancestral process, i.e., in terms of the joint factorial moments of the offspring variables ν1,…,νN.The Kingman coalescent appears in the limit as the population size N tends to infinity if and only if triple mergers are asymptotically negligible in comparison with binary mergers. In this case, a simple upper bound for the rate of convergence of the finite-dimensional distributions is derived. It depends (up to some constants) only on the three factorial moments E((ν1)2), E((ν1)3) and E((ν1)2(ν2)2), where (x)k := x(x-1)…(x-k+1). Examples are the Wright-Fisher model, where the rate of convergence is of order N-1, and the Moran model, with a convergence rate of order N-2.

On the genealogy of populations: trees, branches and offspring

“…This quantity can be interpreted as the time from the common ancestor of the whole population and it has an evident relevance in paleontology. In fact, mtDNA of a single species is only transmitted by female and, therefore, can be considered as an This two quantities can be studied in the context of the coalescence problem which has been widely investigated in a number of papers in the last two decades [2,8,11,12,13,14,17,18,21,22], and is still investigated in present times [3,9,10,19]. We will come back to this approach in next two sections.…”

On the total length of external branches for beta-coalescents

“…Kingman showed in [27] that the ancestral trees of a sample of size n in populations with size N evolving by a Wright-Fisher model will converge weakly to Kingman's n-coalescent for N → ∞ (after a suitable time-change). This result is relatively robust if population evolution deviates from the Wright-Fisher model (see [27] or [28]). However, there is evidence that there are populations where the gene genealogy of a sample is not described well by Kingman's n-coalescent.…”

On the Robustness of the Extension of the One-Third Law of Evolution to the Multi-Player Game

“…This ensures that the probability of a 3-merger, and then the probability of a k-merger for any k ≥ 3 by monotonicity, can be neglected compared to the probability of a 2-merger in the absence of selection when the population size is large enough. Actually this assumption guarantees that the ancestral neutral process in the limit of a large population size with c −1 N time steps as unit of time will be the Kingman [6] coalescent [12]. This assumption holds unless the contributions of individuals from one time step to the next are highly skewed [2,15,17], which is unlikely in most real situations.…”