Second-order asymptotics for the block counting process in a class of regularly varying Λ -coalescents
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Limic, Vlada [1] ; Talarczyk, Anna [2]
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[1] University of Paris-Sud
University of Paris-Sud
Arrondissement de Palaiseau, Francia
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[2] University of Warsaw
University of Warsaw
Warszawa, Polonia
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 3, 2015, págs. 1419-1455
- Idioma: inglés
- DOI: 10.1214/13-AOP902
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Resumen
Consider a standard Λ-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 Nt 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 Nt/vt→1 as t→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 Λ has a density near zero which behaves as x−β with β∈(0,1), then the process (ε−1/(1+β)(Nεt/vεt−1))t≥0 converges in law as ε→0 in the Skorokhod space to a totally skewed (1+β)-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.
