The $\Lambda$-coalescent speed of coming down from infinity

Abstract

Consider a $\Lambda$-coalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number $N_t$ of blocks at any positive time $t>0$). We exhibit a deterministic function $v:(0,\infty )\to (0,\infty )$ such that $N_t/v(t)\to 1$, almost surely, and in $L^p$ for any $p\ge 1$, as $t\to 0$. Our approach relies on a novel martingale technique.

Published in at http://dx.doi.org/10.1214/09-AOP475 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Differential equation approximations for Markov chains
• Small-time behavior of beta coalescents
• Beta-coalescents and continuous stable random trees
• The $\Lambda$-coalescent speed of coming down from infinity
• Global divergence of spatial coalescents

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• The $\Lambda$-coalescent speed of coming down from infinity
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• The number of small blocks in exchangeable random partitions
• Poisson representations of branching Markov and measure-valued branching processes
• A small-time coupling between $\Lambda$-coalescents and branching processes
• Smaller population size at the MRCA time for stationary branching processes
• Second order behavior of the block counting process of beta coalescents
• The tree length of an evolving coalescent
• Diffusion Limits at Small Times for Coalescent Processes with Mutation and Selection
• The equilibrium allele frequency distribution for a population with reproductive skew