Can I read Small-time Behavior of Beta Coalescents on EtoBox?

Small-time Behavior of Beta Coalescents by Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason is a scholarly article available to read on EtoBox.

What is Small-time Behavior of Beta Coalescents about?

For a finite measure $\varLambda$ on $[0,1]$, the $\varLambda$-coalescent is a coalescent process such that, whenever there are $b$ clusters, each $k$-tuple of clusters merges into one at rate $\int_0^1x^{k-2}(1-x)^{b-k}\varLambda(\mathrm{d}x)$. It has recently been shown that if $1<\alpha<2$, the $\varLambda$-coalescent in which $\varLambda$ is the $\operatorname {Beta}(2-\alpha,\alpha)$ distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an $\alpha$-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 $\varLambda$-coalescents for which $\varLambda$ has the same asymptotic behavior near zero as the $\operatorname {Beta}(2-\al

Author

Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason

Published

2006

Language

EN