On the maximal displacement of subcritical branching random walks with or without killing
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913999437692928 |
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| author | Hou, Haojie Zhang, Shuxiong |
| author_facet | Hou, Haojie Zhang, Shuxiong |
| contents | Consider a subcritical branching random walk $\{Z_k\}_{k\geq 0}$ with offspring distribution $\{p_k\}_{k\geq 0}$ and step size $X$. Let $M_n$ denote the rightmost position reached by $\{Z_k\}_{k\geq 0}$ up to generation $n$, and define $M := \sup_{n\geq 0} M_n$. In this paper we give asymptotics of tail probability of $M$ under optimal assumptions $\sum^{\infty}_{k=1}(k\log k) p_k<\infty$ and $\mathbb{E}[Xe^{γX}]<\infty$, where $γ>0$ is a constant such that $\mathbb{E}[e^{γX}]=\frac{1}{m}$ and $m=\sum_{k=0}^\infty kp_k\in (0,1)$. 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 $m\mathbb{E}[X e^{γX}]$ such that
\begin{align*}
\lim_{n\to\infty}e^{γcn}\mathbb{P}(M_n\geq cn)= \left\{
\begin{aligned}
&κ\in(0,1], &c\in\big(0,m\mathbb{E}[Xe^{γX}]\big);
&0, &c\in\big(m\mathbb{E}[Xe^{γX}],\infty\big),
\end{aligned}
\right.
\end{align*}
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15156 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the maximal displacement of subcritical branching random walks with or without killing Hou, Haojie Zhang, Shuxiong Probability Consider a subcritical branching random walk $\{Z_k\}_{k\geq 0}$ with offspring distribution $\{p_k\}_{k\geq 0}$ and step size $X$. Let $M_n$ denote the rightmost position reached by $\{Z_k\}_{k\geq 0}$ up to generation $n$, and define $M := \sup_{n\geq 0} M_n$. In this paper we give asymptotics of tail probability of $M$ under optimal assumptions $\sum^{\infty}_{k=1}(k\log k) p_k<\infty$ and $\mathbb{E}[Xe^{γX}]<\infty$, where $γ>0$ is a constant such that $\mathbb{E}[e^{γX}]=\frac{1}{m}$ and $m=\sum_{k=0}^\infty kp_k\in (0,1)$. 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 $m\mathbb{E}[X e^{γX}]$ such that \begin{align*} \lim_{n\to\infty}e^{γcn}\mathbb{P}(M_n\geq cn)= \left\{ \begin{aligned} &κ\in(0,1], &c\in\big(0,m\mathbb{E}[Xe^{γX}]\big); &0, &c\in\big(m\mathbb{E}[Xe^{γX}],\infty\big), \end{aligned} \right. \end{align*} 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. |
| title | On the maximal displacement of subcritical branching random walks with or without killing |
| topic | Probability |
| url | https://arxiv.org/abs/2508.15156 |