# Small-time behavior of beta coalescents

Julien Berestycki; Nathanaël Berestycki; Jason Schweinsberg

Annales de l'I.H.P. Probabilités et statistiques (2008)

- Volume: 44, Issue: 2, page 214-238
- ISSN: 0246-0203

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topBerestycki, Julien, Berestycki, Nathanaël, and Schweinsberg, Jason. "Small-time behavior of beta coalescents." Annales de l'I.H.P. Probabilités et statistiques 44.2 (2008): 214-238. <http://eudml.org/doc/77967>.

@article{Berestycki2008,

abstract = {For a finite measure Λ on [0, 1], the Λ-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate ∫01xk−2(1−x)b−kΛ(dx). It has recently been shown that if 1<α<2, the Λ-coalescent in which Λ is the Beta (2−α, α) distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an α-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time asymptotics of beta coalescents. We prove an a.s. limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents, and we find the sum of the lengths of the branches in the coalescent tree, both of which are determined by the behavior of coalescents at small times. We extend most of these results to other Λ-coalescents for which Λ has the same asymptotic behavior near zero as the Beta (2−α, α) distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study small-time properties of Λ-coalescents.},

author = {Berestycki, Julien, Berestycki, Nathanaël, Schweinsberg, Jason},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {coalescence; continuous-state branching process; coalescent with multiple mergers},

language = {eng},

number = {2},

pages = {214-238},

publisher = {Gauthier-Villars},

title = {Small-time behavior of beta coalescents},

url = {http://eudml.org/doc/77967},

volume = {44},

year = {2008},

}

TY - JOUR

AU - Berestycki, Julien

AU - Berestycki, Nathanaël

AU - Schweinsberg, Jason

TI - Small-time behavior of beta coalescents

JO - Annales de l'I.H.P. Probabilités et statistiques

PY - 2008

PB - Gauthier-Villars

VL - 44

IS - 2

SP - 214

EP - 238

AB - For a finite measure Λ on [0, 1], the Λ-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate ∫01xk−2(1−x)b−kΛ(dx). It has recently been shown that if 1<α<2, the Λ-coalescent in which Λ is the Beta (2−α, α) distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an α-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time asymptotics of beta coalescents. We prove an a.s. limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents, and we find the sum of the lengths of the branches in the coalescent tree, both of which are determined by the behavior of coalescents at small times. We extend most of these results to other Λ-coalescents for which Λ has the same asymptotic behavior near zero as the Beta (2−α, α) distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study small-time properties of Λ-coalescents.

LA - eng

KW - coalescence; continuous-state branching process; coalescent with multiple mergers

UR - http://eudml.org/doc/77967

ER -

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