On the external branches of coalescents with multiple collisions
نویسندگان
چکیده
A recursion for the joint moments of the external branch lengths for coalescents with multiple collisions (Λ-coalescents) is provided. This recursion is used to derive asymptotic results as the sample size n tends to infinity for the joint moments of the external branch lengths and for the moments of the total external branch length of the Bolthausen–Sznitman coalescent. These asymptotic results are based on a differential equation approach, which is as well useful to obtain exact solutions for the joint moments of the external branch lengths for the Bolthausen– Sznitman coalescent. The results for example show that the lengths of two randomly chosen external branches are positively correlated for the Bolthausen–Sznitman coalescent, whereas they are negatively correlated for the Kingman coalescent provided that n ≥ 4.
منابع مشابه
On Asymptotics of Exchangeable Coalescents with Multiple Collisions
We study the number of collisions, Xn, of an exchangeable coalescent with multiple collisions ( -coalescent)which startswithn particles and is driven by rates determined by a finite characteristic measure ν(dx) = x−2 (dx). Via a coupling technique, we derive limiting laws of Xn, using previous results on regenerative compositions derived from stick-breaking partitions of the unit interval. The ...
متن کاملConditions for exchangeable coalescents to come down from infinity
An improved condition to come down from infinity for exchangeable coalescent processes with simultaneous multiple collisions of ancestral lineages is provided. For non-critical coalescents this leads to an improved necessary and sufficient condition for the coalescent to come down from infinity. An analog conjecture for the full class of exchangeable coalescents is presented. New examples of cr...
متن کاملOn the Number of Allelic Types for Samples Taken from Exchangeable Coalescents with Mutation
Exchangeable coalescents are Markovian processes with state space E , the set of equivalence relations (partitions) on N := {1, 2, . . .} with a block merging mechanism. The class of exchangeable coalescents with multiple collisions has been independently introduced by Pitman [23] and Sagitov [24]. These processes can be characterized by a finite measure Λ on the unit interval [0, 1] and are he...
متن کاملAlmost 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....
متن کاملDiffusion limits at small times for Λ - coalescents with a Kingman component *
We consider standard Λ-coalescents (or coalescents with multiple collisions) with a non-trivial “Kingman part”. Equivalently, the driving measure Λ has an atom at 0; Λ({0}) = c > 0. It is known that all such coalescents come down from infinity. Moreover, the number of blocks Nt is asymptotic to v(t) = 2/(ct) as t → 0. In the present paper we investigate the second-order asymptotics of Nt in the...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2012