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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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

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

First available in Project Euclid: 5 May 2015

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Zentralblatt MATH identifier

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

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


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.

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  • [1] Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 3–48.
  • [2] Berestycki, J., Berestycki, N. and Limic, V. (2014). A small-time coupling between $\Lambda$-coalescents and branching processes. Ann. Appl. Probab. 24 449–475.
  • [3] Berestycki, J., Berestycki, N. and Limic, V. (2010). The $\Lambda$-coalescent speed of coming down from infinity. Ann. Probab. 38 207–233.
  • [4] Berestycki, N. (2009). Recent Progress in Coalescent Theory. Ensaios Matemáticos 16. Sociedade Brasileira de Matemática, Rio de Janeiro.
  • [5] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.
  • [6] Bertoin, J. and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 147–181 (electronic).
  • [7] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [8] Birkner, M., Blath, J., Capaldo, M., Etheridge, A., Möhle, M., Schweinsberg, J. and Wakolbinger, A. (2005). Alpha-stable branching and beta-coalescents. Electron. J. Probab. 10 303–325 (electronic).
  • [9] Bojdecki, T., Gorostiza, L. G. and Talarczyk, A. (2007). A long range dependence stable process and an infinite variance branching system. Ann. Probab. 35 500–527.
  • [10] Donnelly, P. and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Probab. 27 166–205.
  • [11] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [12] Iscoe, I. (1986). A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Related Fields 71 85–116.
  • [13] Limic, V. (2011). Processus de Coalescence et Marches Aléatoires Renforcées : Un guide à travers martingales et couplage. Habilitation thesis (in French and English). Available at
  • [14] Limic, V. (2010). On the speed of coming down from infinity for $\Xi$-coalescent processes. Electron. J. Probab. 15 217–240.
  • [15] Limic, V. (2012). Genealogies of regular exchangeable coalescents with applications to sampling. Ann. Inst. Henri Poincaré Probab. Stat. 48 706–720.
  • [16] Peszat, S. and Zabczyk, J. (2007). Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach. Encyclopedia of Mathematics and Its Applications 113. Cambridge Univ. Press, Cambridge.
  • [17] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.
  • [18] Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 1116–1125.
  • [19] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York.
  • [20] Schweinsberg, J. (2000). A necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity. Electron. Commun. Probab. 5 1–11 (electronic).
  • [21] Schweinsberg, J. (2000). Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5 50 pp. (electronic).
  • [22] Schweinsberg, J. (2003). Coalescent processes obtained from supercritical Galton–Watson processes. Stochastic Process. Appl. 106 107–139.
  • [23] Schweinsberg, J. (2012). Dynamics of the evolving Bolthausen–Sznitman coalecent. Electron. J. Probab. 17 50.