On the maximal displacement of subcritical branching random walks with or without killing

2025/08/21 by Haojie Hou, Hou, Haojie, Shuxiong Zhang +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2508.15156

openalex publication_date 2025/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider a subcritical branching random walk \Z_k\_{k≥ 0} with offspring distribution \p_k\_{k≥ 0} and step size X. Let M_n denote the rightmost position reached by \Z_k\_{k≥ 0} up to generation n, and define M := sup_{n≥ 0} M_n. In this paper we give asymptotics of tail probability of M under optimal assumptions ∑^∞_k=1(klog k) p_k<∞ and 𝔼[Xe^γX]<∞, where γ>0 is a constant such that 𝔼[e^γX]=(1)/(m) and m=∑_k=0^∞ kp_k∈ (0,1). Moreover, we confirm the conjecture of Neuman and Zheng [Probab. Theory Related Fields. 167 (2017) 1137--1164] by establishing the existence of a critical value m𝔼[X e^γX] such that lim_n→∞e^γcnℙ(M_n≥ cn)= \ \beginaligned amp;κ∈(0,1], amp;c∈(0,m𝔼[Xe^γX]); amp;0, amp;c∈(m𝔼[Xe^γX],∞), \endaligned . where κ represents the non-zero limit. Finally, we extend these results to the maximal displacement of branching random walks with killing. Interestingly, this limit can be characterized through both the global minimum of a random walk with positive drift and the maximal displacement of the branching random walk without killing.

Citations

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• On the maximal displacement of critical branching random walk in random environment