SECOND-ORDER ASYMPTOTICS FOR THE BLOCK COUNTING PROCESS IN A CLASS OF REGULARLY VARYING Λ-COALESCENTS

• 成果类型: Article
• 署名作者: Limic, Vlada; Talarczyk, Anna
• 署名单位: Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); University of Warsaw
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP902
• 发表日期: 2015
• 页码: 1419-1455
• 关键词: speed; time

摘要

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) -> 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 Lambda has a density near zero which behaves as x(-beta) with beta is an element of (0, 1), then the process (epsilon(-1/(1 + beta))(N-epsilon t/v(epsilon t) - 1))(t >= 0) converges in law as epsilon -> 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 Levy noise.