A Construction of a β-Coalescent via the Pruning of Binary Trees

Abstract

Abstract. Considering a random binary tree with n labelled leaves, we use a pruning procedure on this tree in order to construct a β( 3 2 , 1 2 )-coalescent process. We also use the continuous analogue of this construction, i.e. a pruning procedure on Aldous's continuum random tree, to construct a continuous state space process that has the same structure as the β-coalescent process up to some time change. These two constructions enable us to obtain results on the coalescent process such as the asymptotics on the number of coalescent events or the law of the blocks involved in the last coalescent event.

Λ-coalescents: a survey

“…For a suitable choice of the offspring distribution, the dynamics reproduce the symmetric beta coalescent with 1 2 ≤ α < 1. The case α = 1 2 was represented by random binary tree cutting in [2]. The last collision in the symmetric beta case was studied in [33].…”

“…The constants m and s 2 given in Table 2 are defined by n(α − 1) ( α− 1)n 1/α α-stable [16,24,30] a = 1 b > 0 n(log n) −1 + n (log n) 2 1-stable [19,36] (b = 1), [30] n(log n) −2 log log n 1 < a < 2 b > 0 0 α −1 (α)n α ∞ 0 e −αS(t) dt [27,32] Table 4: Limit distributions for (M n − g n )/ h n for beta(a, b)-coalescents. Here ρ is the rate of a Poisson process on the coalescent tree, a 2 = 2 − √ 2, the other constants c 1 , c 2 , α, and β are the same as in Table 3, and ζ is the sum of a centered normal random variable with variance θc 1 and ρc 2 (β) −α −1 times an independent α-stable random variable.…”

“…Here ρ is the rate of a Poisson process on the coalescent tree, a 2 = 2 − √ 2, the other constants c 1 , c 2 , α, and β are the same as in Table 3, and ζ is the sum of a centered normal random variable with variance θc 1 and ρc 2 (β) −α −1 times an independent α-stable random variable. [38] a = 1 b > 0 n(b log n) −1 + ρn b(log n) 2 1-stable [18] (b = 1), [30] b −1 n log log n(log n) −2 0 < a < 1 b = 2 − a Unknown a > 1 b > 0 0 ρn ∞ 0 e −S t dt [45] The limit distribution for X n in the Bolthausen-Sznitman coalescent was obtained in [19] with the aid of the singularity analysis of generating functions, and later probabilistically via coupling with random walks with barrier in [36] by using a relation to random recursive trees; see Section 4.2. The latter approach proved useful [35] to study collisions in beta(a, 1)-coalescents with 0 < a < 2.…”

“…The constants are c 1= (α + 1)(α − 1)/(2 − α), c 2 = (α + 1)(α − 1) 1+α −1 / α −1 (2 − α), a 3 = 1 2 (3 − √ 5 ), α = 2 − a, and β = 1 + α − α 2 , and η is some nondegenerate absolutely continuous random variable. log n) α −1 α-stable[38] a 3 < a < 1 b = 2 − a c 1 n a c 2 (β −1 n β ) α −1 α-stable [38] a = 1 b > 0 n(b log n) −1 + n b(log n)2 1-stable[18] (b = 1),[30] b −1 n log log n(log n) = a + b − 1 (a − 1)(2 − a) (2[ (a + b − 1) − (b)] − (a + b − 2)[[ (a + b − 1) − (b)] 2 + (b) − (a + b − 1)]),at https://www.cambridge.org/core/terms. https://doi.org/10.1239/jap/1417528464…”

Λ-coalescents: a survey

“…The constants m and s 2 given in Table 2 are defined by ( α− 1)n 1/α α-stable [16,24,30] a = 1 b > 0 n(log n) −1 + n (log n) 2 1-stable [19,36] (b = 1), [30] n(log n) −2 log log n 1 < a < 2 b > 0 0 α −1 (α)n α Here ρ is the rate of a Poisson process on the coalescent tree, a 2 = 2 − √ 2, the other constants c 1 , c 2 , α, and β are the same as in Table 3, and ζ is the sum of a centered normal random variable with variance θc 1 and ρc 2 (β) −α −1 times an independent α-stable random variable. The limit distribution for X n in the Bolthausen-Sznitman coalescent was obtained in [19] with the aid of the singularity analysis of generating functions, and later probabilistically via coupling with random walks with barrier in [36] by using a relation to random recursive trees; see Section 4.2.…”