@article{70daa8d3d1c0475bbb54af2f85288fdf,
title = "Tail probability of maximal displacement in critical branching L{\'e}vy process with stable branching",
abstract = "Consider a critical branching L{\'e}vy process {Xt,t ≥ 0} with branching rate β>0, offspring distribution {pk: k ≥ 0} and spatial motion {ξt, Px}.Foranyt ≥ 0, let Nt be the collection of particles alive at time t, and, for any u ∈ Nt,letXu (t) be the position of u at time t. We study the tail probability of the maximal displacement M:= supt>0 supu∈Nt Xu (t) under the assumption lim (Formula presented) for some α ∈(1,2), (Formula presented) for some r > 2α/(α − 1). Our main result is a generalization of the main result of Sawyer and Fleischman (1979) for branching Brownian motions and that of Lalley and Shao (2015) for branching random walks, both of these results are proved under the assumption (Formula presented).",
keywords = "Branching L{\'e}vy process, Feynman-Kac representation, critical branching process",
author = "Haojie Hou and Yiyang Jiang and Ren, {Yan Xia} and Renming Song",
note = "We thank the referees for many helpful suggestions, particularly for suggesting the strengthened version of Lemma 2.1 and its streamlined proof. Part of the research of this paper was carried out while the fourth-named author was visiting Jiangsu Normal University, where he was partially supported by a grant from the Natural Science Foundation of China (11931004, Yingchao Xie). The research of this project is supported in part by the National Key R&D Program of China (No. 2020YFA0712900). The third-named author was supported by NSFC (Grant Nos. 12071011 and 12231002) and The Fundamental Research Funds for the Central Universities, Peking University LMEQF. The fourth-named author was supported in part by a grant from the Simons Foundation (#960480, Renming Song). The research of this project is supported in part by the National Key R&D Program of China (No. 2020YFA0712900). The third-named author was supported by NSFC (Grant Nos. 12071011 and 12231002) and The Fundamental Research Funds for the Central Universities, Peking University LMEQF. The fourth-named author was supported in part by a grant from the Simons Foundation (#960480, Renming Song). We thank the referees for many helpful suggestions, particularly for suggesting the strengthened version of Lemma 2.1 and its streamlined proof. Part of the research of this paper was carried out while the fourth-named author was visiting Jiangsu Normal University, where he was partially supported by a grant from the Natural Science Foundation of China (11931004, Yingchao Xie).",
year = "2025",
month = feb,
doi = "10.3150/24-BEJ1742",
language = "English (US)",
volume = "31",
pages = "630--648",
journal = "Bernoulli",
issn = "1350-7265",
publisher = "Bernoulli Society for Mathematical Statistics and Probability",
number = "1",
}