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Scaling limits of branching random walks and branching-stable processes

Part of: Limit theorems Markov processes

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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