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Scaling limits of branching random walks and branching-stable processes

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  04 August 2022

Jean Bertoin  and
Hairuo Yang
Jean Bertoin*
Affiliation:
University of Zurich
Hairuo Yang*
Affiliation:
University of Zurich
*
*Postal address: Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland.
*Postal address: Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland.
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Abstract

Branching-stable processes have recently appeared as counterparts of stable subordinators, when addition of real variables is replaced by branching mechanisms for point processes. Here we are interested in their domains of attraction and describe explicit conditions for a branching random walk to converge after a proper magnification to a branching-stable process. This contrasts with deep results obtained during the past decade on the asymptotic behavior of branching random walks and which involve either shifting without rescaling, or demagnification.

Keywords

Branching random walkscaling limitbranching-stable processes

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 1009 - 1025
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Aïdékon, E. (2013). Convergence in law of the minimum of a branching random walk. Ann. Prob. 41, 1362–1426.CrossRefGoogle Scholar
Aïdékon, E., Berestycki, J., Brunet, E. and Shi, Z. (2013). Branching Brownian motion seen from its tip. Prob. Theory Related Fields 157, 405–451.CrossRefGoogle Scholar
Arguin, L.-P., Bovier, A. and Kistler, N. (2012). Poissonian statistics in the extremal process of branching Brownian motion. Ann. Appl. Prob. 22, 1693–1711.CrossRefGoogle Scholar
Arguin, L.-P., Bovier, A. and Kistler, N. (2013). The extremal process of branching Brownian motion. Prob. Theory Related Fields 157, 535–574.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover Publications.Google Scholar
Bertoin, J. and Mallein, B. (2019). Infinitely ramified point measures and branching Lévy processes. Ann. Prob. 47, 1619–1652.CrossRefGoogle Scholar
Bertoin, J., Cortines, A. and Mallein, B. (2018). Branching-stable point measures and processes. Adv. Appl. Prob. 50, 1294–1314.CrossRefGoogle Scholar
Bhattacharya, A., Hazra, R. S. and Roy, P. (2017). Point process convergence for branching random walks with regularly varying steps. Ann. Inst. H. Poincaré Prob. Statist. 53, 802–818.CrossRefGoogle Scholar
Bhattacharya, A., Hazra, R. S. and Roy, P. (2018). Branching random walks, stable point processes and regular variation. Stoch. Process. Appl. 128, 182–210.CrossRefGoogle Scholar
Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25–37.CrossRefGoogle Scholar
Biggins, J. D. (1992). Uniform convergence of martingales in the branching random walk. Ann. Prob. 20, 137–151.CrossRefGoogle Scholar
Bramson, M. D. (1978). Maximal displacement of branching Brownian motion. Commun. Pure Appl. Math. 31, 531–581.CrossRefGoogle Scholar
Bramson, M. D. (1983). Convergence of Solutions of the Kolmogorov Equation to Travelling Waves (Mem. Amer. Math. Soc. 44). American Mathematical Society.Google Scholar
Durrett, R. (1983). Maxima of branching random walks. Z. Wahrscheinlichkeitsth. 62, 165–170.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence (Wiley Series in Probability and Mathematical Statistics). John Wiley.CrossRefGoogle Scholar
Gnedenko, B. V. and Kolmogorov, A. N. (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, MA.Google Scholar
Hult, H. and Lindskog, F. (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. N.S. 80, 121–140.CrossRefGoogle Scholar
Ibragimov, I. A. and Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen.Google Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes (Grundlehren der Mathematischen Wissenschaften 288), 2nd edn. Springer, Berlin.Google Scholar
Jagers, P. (1989). General branching processes as Markov fields. Stoch. Process. Appl. 32, 183–212.CrossRefGoogle Scholar
Kallenberg, O. (2002). Foundations of Modern Probability (Probability and its Applications (New York)), 2nd edn. Springer, New York.Google Scholar
Lindskog, F., Resnick, S. I. and Roy, J. (2014). Regularly varying measures on metric spaces: hidden regular variation and hidden jumps. Prob. Surv. 11, 270–314.CrossRefGoogle Scholar
Madaule, T. (2017). Convergence in law for the branching random walk seen from its tip. J. Theoret. Prob. 30, 27–63.CrossRefGoogle Scholar
Resnick, S. I. (2007). Heavy-tail Phenomena: Probabilistic and Statistical Modeling (Springer Series in Operations Research and Financial Engineering). Springer, New York.Google Scholar
Shi, Z. (2015). Branching Random Walks: Lecture Notes From the 42nd Probability Summer School, Saint Flour 2012 (Lecture Notes Math. 2151). Springer, Cham.Google Scholar
Uchiyama, K. (1982). Spatial growth of a branching process of particles living in ${\bf R}^{d}$ . Ann. Prob. 10, 896–918.CrossRefGoogle Scholar