Second-order asymptotics for the block counting process in a class of regularly varying $\Lambda$-coalescents

Abstract

Consider a standard $\Lambda$-coalescent that comes down from infinity. Such a coalescent starts from a configuration consisting of infinitely many blocks at time $0$, but its number of blocks $N_t$ is a finite random variable at each positive time $t$. Berestycki et al. [Ann. Probab. 38 (2010) 207-233] found the first-order approximation $v$ for the process $N$ at small times. This is a deterministic function satisfying $N_t/v_t\to 1$ as $t\to 0$. The present paper reports on the first progress in the study of the second-order asymptotics for $N$ at small times. We show that, if the driving measure $\Lambda$ has a density near zero which behaves as $x^{-\beta }$ with $\beta \in (0,1)$, then the process $({\epsilon }^{-1/(1+\beta )}(N_{\epsilon t}/v_{\epsilon t}-1){)}_{t\ge 0}$ converges in law as $\epsilon \to 0$ in the Skorokhod space to a totally skewed $(1+\beta )$-stable process. Moreover, this process is a unique solution of a related stochastic differential equation of Ornstein-Uhlenbeck type, with a completely asymmetric stable Lévy noise.

Published at http://dx.doi.org/10.1214/13-AOP902 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

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

Cited by in corpus (7)

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• Diffusion Limits at Small Times for Coalescent Processes with Mutation and Selection
• Diffusion limits at small times for coalescents with a Kingman component
• Integral representation of probabilities in Kingman coalescent
• The joint fluctuations of the lengths of the Beta$(2-\alpha ,\alpha )$-coalescents
• On the boundary classification of $\Lambda$-Wright-Fisher processes with frequency-dependent selection