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May 2015 Second-order asymptotics for the block counting process in a class of regularly varying ${\Lambda}$-coalescents
Vlada Limic, Anna Talarczyk
Ann. Probab. 43(3): 1419-1455 (May 2015). DOI: 10.1214/13-AOP902

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 $N_{t}/v_{t}\to1$ as $t\to0$. 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 $(\varepsilon^{-1/(1+\beta)}(N_{\varepsilon t}/v_{\varepsilon t}-1))_{t\ge0}$ converges in law as $\varepsilon\to0$ 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.

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Vlada Limic. Anna Talarczyk. "Second-order asymptotics for the block counting process in a class of regularly varying ${\Lambda}$-coalescents." Ann. Probab. 43 (3) 1419 - 1455, May 2015. https://doi.org/10.1214/13-AOP902

Information

Published: May 2015
First available in Project Euclid: 5 May 2015

zbMATH: 1327.60143
MathSciNet: MR3342667
Digital Object Identifier: 10.1214/13-AOP902

Subjects:
Primary: 60J25
Secondary: 60F17 , 60G52 , 60G55 , 92D25

Keywords: ${\Lambda }$-coalescent , coming down from infinity , Ornstein–Uhlenbeck process , Poisson random measure , second-order approximations , Stable Lévy process

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 3 • May 2015
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