Small-time behavior of beta coalescents
By J. Berestycki, Nathanaël Berestycki, Jason Schweinsberg. 2006
84 citations recorded by the source corpus.
For a finite measureon (0,1), the �-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate R 1 0 x k 2 (1 x) b k �(dx). It has recently been shown that if 1 < � < 2, the �-coalescent in whichis the Beta(2 �,�) distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an �-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 �-coalescents for whichhas the same asymptotic behavior near zero as the Beta(2 �,�) distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study small-time properties of �-coalescents.
Subfields: Condensed Matter Physics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/54486a6383605d0c53c89e2fa0b27275387cafa9
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In Condensed Matter Physics
In Mathematical Physics
- Lancelot F. James
- A. Winter
- Jim Pitman
- Nathanaël Berestycki
- Peter Pfaffelhuber
- Andrea Greven
- J. Berestycki
- V. Limic
Citations in the corpus
- Recent progress in coalescent theory (2009, unknown): “is the most useful for capturing fine aspects of the small-time behaviour – although see [19] for what you can do with just the lookdown process).”
- The Λ-coalescent speed of coming down from infinity (2008, background): “The speed of CDI was recently determined for Beta-coalescents and their “perturbations” in Berestycki, Berestycki and Schweinsberg [4] and [5] and Bertoin and Le Gall [6] (where convergence is established in probability).”
- Lamperti-type laws (2007, methodology): “However, the intensity νt, which plays a fundamental role in (6; 9) is only known up to its Laplace transform.”
- Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees) (2009, background): “Such coalescent processes have since been the subject of many papers (see, for example, [3,5,6,21,23]).”