On Asymptotics of Exchangeable Coalescents with Multiple Collisions

Gnedin, Alexander; Iksanov, Alexander; Möhle, Martin

doi:10.1017/s0021900200005064

Cited by 22 publications

(51 citation statements)

References 25 publications

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“…We let K (i) n denote this random variable when the n initial articles are in environment i. When there is a unique environment (κ = 1), this question has been treated by several authors [16,15,24,23,18,14]. In a varying environment, we obtain as a direct consequence of our results: Theorem 6.1.…”

Section: Number Of Collisionsmentioning

confidence: 65%

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Bivariate Markov chains converging to Lamperti transform Markov additive processes

Haas

Stephenson

2018

Stochastic Processes and their Applications

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Motivated by various applications, we describe the scaling limits of bivariate Markov chains (X, J) on Z + × {1, . . . , κ} where X can be viewed as a position marginal and {1, . . . , κ} is a set of κ types. The chain starts from an initial value (n, i) ∈ Z + × {1, . . . , κ}, with i fixed and n → ∞, and typically we will assume that the macroscopic jumps of the marginal X are rare, i.e. arrive with a probability proportional to a negative power of the current state. We also assume that X is non-increasing. We then observe different asymptotic regimes according to whether the rate of type change is proportional to, faster than, or slower than the macroscopic jump rate. In these different situations, we obtain in the scaling limit Lamperti transforms of Markov additive processes, that sometimes reduce to standard positive self-similar Markov processes. As first examples of applications, we study the number of collisions in coalescents in varying environment and the scaling limits of Markov random walks with a barrier. This completes previous results obtained by Haas and Miermont [18] and Bertoin and Kortchemski [9] in the monotype setting. In a companion paper, we will use these results as a building block to study the scaling limits of multi-type Markov branching trees, with applications to growing models of random trees and multi-type Galton-Watson trees.

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Section: Number Of Collisionsmentioning

confidence: 65%

“…Since Theorem 3.1 is already proven, the only thing we have to check is that σ (i) = I (i) a.s., with I (i) the extinction time of X (i) . Indeed, if this holds for all converging subsequences, this will (1) imply the convergence in distribution of (15) to (X (i) , Z (i) , I (i) )…”

Section: Proof Of Theorem 32: Scaling Limit Of the Absorption Timementioning

confidence: 99%

Bivariate Markov chains converging to Lamperti transform Markov additive processes

Haas

Stephenson

2018

Stochastic Processes and their Applications

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“…For simple Λ-coalescents each partition Π ∞ (t) has a dust component. Therefore, a coupling between the infinite coalescent and a subordinator (which is a compound Poisson process in the case m −2 < ∞) [16,19] can be applied to relate N n with a simpler counting process derived from the dust component. We briefly summarise the combinatorial part of this connection.…”

Section: Simple Coalescents: Functional Limitsmentioning

confidence: 99%

“…Among the functionals characterising the speed of coalescence, the total number of collisions X n (transitions of Π n until absorption) has attracted the most attention, see [16,19,23,24,26] and a survey paper [17]. Here, we are interested in more delicate properties of the coalescent process by distinguishing mergers of various sizes, that is decomposing the total number of collisions as X n = n k=2 X n,k , 2 ≤ k ≤ n, where X n,k is the number of collisions resulting in a merger of k blocks.…”

Section: Introductionmentioning

confidence: 99%

The collision spectrum of $\Lambda$-coalescents

Gnedin¹,

Iksanov²,

Marynych³

et al. 2018

Ann. Appl. Probab.

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Λ-coalescents model the evolution of a coalescing system in which any number of blocks randomly sampled from the whole may merge into a larger block. For the coalescent restricted to initially n singletons we study the collision spectrum (X n,k : 2 ≤ k ≤ n), where X n,k counts, throughout the history of the process, the number of collisions involving exactly k blocks. Our focus is on the large n asymptotics of the joint distribution of the X n,k 's, as well as on functional limits for the bulk of the spectrum for simple coalescents. Similarly to the previous studies of the total number of collisions, the asymptotics of the collision spectrum largely depends on the behaviour of the measure Λ in the vicinity of 0. In particular, for beta(a, b)-coalescents different types of limit distributions occur depending on whether 0 < a ≤ 1, 1 < a < 2, a = 2 or a > 2.

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“…In this case, the process Π (n) can be constructed using a subordinator. This construction can be found on page 7 of [19] and the original idea is in [27]. Let ν(dx) = x −2 Λ(dx) and ν be the push-forward of ν by the transformation x → − ln(1 − x).…”

Section: Let Us Consider the Case Of Coalescents Satisfyingmentioning

confidence: 99%

Asymptotics of the Minimal Clade Size and Related Functionals of Certain Beta-Coalescents

Siri‐Jégousse

Yuan

2015

Acta Appl Math

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This article shows the asymptotics of distributions of various functionals of the Beta(2 − α, α) n-coalescent process with 1 < α < 2 when n goes to infinity. This process is a Markov process taking values in the set of partitions of {1, . . . , n}, evolving from the intial value {1}, · · · , {n} by merging (coalescing) blocks together into one and finally reaching the absorbing state {1, . . . , n}. The minimal clade of 1 is the block which contains 1 at the time of coalescence of the singleton {1}. The limit size of the minimal clade of 1 is provided. To this, we express it as a function of the coalescence time of {1} and sizes of blocks at that time. Another quantity concerning the size of the largest block (at deterministic small time and at the coalescence time of {1}) is also studied.Date: July 25, 2018. 2010 Mathematics Subject Classification. 60J25, 60F05, 92D15. Key words and phrases. Beta-coalescent, Kingman's paintbox construction, minimal clade size, external branch lengths, largest bloc size, block-counting process.Linglong Yuan benefited from the support of the "Agence Nationale de la Recherche": ANR MANEGE (ANR-09-BLAN-0215).

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On Asymptotics of Exchangeable Coalescents with Multiple Collisions

Cited by 22 publications

References 25 publications

Bivariate Markov chains converging to Lamperti transform Markov additive processes

Bivariate Markov chains converging to Lamperti transform Markov additive processes

The collision spectrum of $\Lambda$-coalescents

Asymptotics of the Minimal Clade Size and Related Functionals of Certain Beta-Coalescents

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