On the maximal displacement of near-critical branching random walks
File(s)
OA Location
Author(s)
Neumann, Eyal
Xinghua, Zheng
Type
Journal Article
Abstract
We consider a branching random walk on Z started by n particles at the origin, where each particle disperses according to a mean-zero random walk with bounded support and reproduces with mean number of offspring 1+θ/n. For t≥0, we study Mnt, the rightmost position reached by the branching random walk up to generation [nt]. Under certain moment assumptions on the branching law, we prove that Mnt/n−−√ converges weakly to the rightmost support point of the local time of the limiting super-Brownian motion. The convergence result establishes a sharp exponential decay of the tail distribution of Mnt. We also confirm that when θ>0, the support of the branching random walk grows in a linear speed that is identical to that of the limiting super-Brownian motion which was studied by Pinsky (Ann Probab 23(4):1748–1754, 1995). The rightmost position over all generations, M:=suptMnt, is also shown to converge weakly to that of the limiting super-Brownian motion, whose tail is found to decay like a Gumbel distribution when θ<0.
Date Issued
2021-03-20
Date Acceptance
2021-03-09
Citation
Probability Theory and Related Fields, 2021, 180, pp.199-232
ISSN
0178-8051
Publisher
Springer
Start Page
199
End Page
232
Journal / Book Title
Probability Theory and Related Fields
Volume
180
Copyright Statement
© The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
License URL
Identifier
https://link.springer.com/article/10.1007%2Fs00440-021-01042-8
Subjects
Science & Technology
Physical Sciences
Statistics & Probability
Mathematics
Statistics & Probability
0101 Pure Mathematics
0102 Applied Mathematics
0104 Statistics
Publication Status
Published
Date Publish Online
2021-03-20