The Annals of Probability

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

Vlada Limic and Anna Talarczyk

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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.

Article information

Source
Ann. Probab., Volume 43, Number 3 (2015), 1419-1455.

Dates
First available in Project Euclid: 5 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1430830286

Digital Object Identifier
doi:10.1214/13-AOP902

Mathematical Reviews number (MathSciNet)
MR3342667

Zentralblatt MATH identifier
1327.60143

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60F17: Functional limit theorems; invariance principles 92D25: Population dynamics (general) 60G52: Stable processes 60G55: Point processes

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

Citation

Limic, Vlada; Talarczyk, Anna. Second-order asymptotics for the block counting process in a class of regularly varying ${\Lambda}$-coalescents. Ann. Probab. 43 (2015), no. 3, 1419--1455. doi:10.1214/13-AOP902. https://projecteuclid.org/euclid.aop/1430830286


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