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

Consider a subcritical branching random walk $\{Z_k{\}}_{k\ge 0}$ with offspring distribution $\{p_k{\}}_{k\ge 0}$ and step size $X$. Let $M_n$ denote the rightmost position reached by $\{Z_k{\}}_{k\ge 0}$ up to generation $n$, and define $M:={\sup }_{n\ge 0}{M}_n$. In this paper we give asymptotics of tail probability of $M$ under optimal assumptions ${\sum }_{k=1}^{\mathrm{\infty }}(k\log k)p_k<\mathrm{\infty }$ and $\mathbb{E}[X{e}^{\gamma X}]<\mathrm{\infty }$, where $\gamma >0$ is a constant such that $\mathbb{E}[e^{\gamma X}]=\frac{1}{m}$ and $m={\sum }_{k=0}^{\mathrm{\infty }}k{p}_k\in (0,1)$. 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^{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 $\kappa$ 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.