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

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 $\mathbb{E}[Xe^{\gamma X}]<\infty$, where $\gamma >0$ is a constant such that $\mathbb{E}[e^{\gamma X}]=\frac{1}{m}$ 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 $m\mathbb{E}[X e^{\gamma X}]$ such that \begin{align*} \lim_{n\to\infty}e^{\gamma cn}\mathbb{P}(M_n\geq cn)= \left\{ \begin{aligned} &\kappa \in(0,1], &c\in\big(0,m\mathbb{E}[Xe^{\gamma X}]\big); &0, &c\in\big(m\mathbb{E}[Xe^{\gamma X}],\infty\big), \end{aligned} \right. \end{align*} where $\kappa$ 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.