Small-time behavior of beta coalescents

For a finite measure Lambda on [0, 1], the A-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate integral(1)(0) x(k-2)(1 - x)(b-k) Lambda(dx). It has recently been shown that if 1 < alpha < 2, the Lambda-coalescent in which A is the 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 Lambda-coalescents for which Lambda has the same asymptotic behavior near zero as the Beta(2 - alpha, alpha) distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study small-time properties of Lambda-coalescents.