Small-time behavior of beta coalescents

Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 2, pp. 214-238.

L’objet de ce travail est l’étude du comportement asymptotique en temps petit des Beta-coalescents. Ces processus décrivent la limite d’échelle de la généalogie d’un certain nombre de modèles en génétique des populations. Nous donnons en particulier un théorème de convergence presque sûre pour le nombre de blocs renormalisé. Nous décrivons également le comportement asymptotique des tailles des blocs. Ces résultats permettent de calculer la dimension de Hausdorff et la dimension de packing d’un espace métrique associé à ce type de coalescents, ainsi que la longueur totale des branches de l’arbre de coalescence. Ce dernier résultat correspond à une question qui se pose en génétique des populations. Enfin, ces résultats sont en partie étendus par des arguments de couplage aux cas de $\Lambda$-coalescents pour lesquels la mesure $\Lambda$ a un comportement près de 0 semblable à celui d’une distribution Beta. Les méthodes employées reposent essentiellement sur un lien entre Beta-coalescent et les processus de branchement à espace d’état continu.

For a finite measure $\Lambda$ on $\left[0,1\right]$, the $\Lambda$-coalescent is a coalescent process such that, whenever there are $b$ clusters, each $k$-tuple of clusters merges into one at rate ${\int }_0^1{x}^{k-2(1-x)}b-k\Lambda \left(\mathrm{d}x\right)$. It has recently been shown that if $1<\alpha <2$, the $\Lambda$-coalescent in which $\Lambda$ is the $\mathrm{Beta}(2-\alpha ,\alpha )$ 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 $\mathrm{Beta}(2-\alpha ,\alpha )$ distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study small-time properties of $\Lambda$-coalescents.

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