On the maximal displacement of subcritical branching random walks

Abstract

We study the maximal displacement of a one dimensional subcritical branching random walk initiated by a single particle at the origin. For each $n\in \mathbb {N},$ let $M_{n}$ be the rightmost position reached by the branching random walk up to generation n. Under the assumption that the offspring distribution has a finite third moment and the jump distribution has mean zero and a finite probability generating function, we show that there exists $\rho >1$ such that the function

$$\begin{aligned} g(c,n):=\rho ^{cn} P(M_{n}\ge cn), \quad \hbox {for each }c>0 \hbox { and } n\in \mathbb {N}, \end{aligned}$$

satisfies the following properties: there exist $0<\underline{\delta }\le \overline{\delta } < {\infty }$ such that if $c<\underline{\delta }$, then

$$\begin{aligned} 0<\liminf _{n\rightarrow \infty } g (c,n)\le \limsup _{n\rightarrow \infty } g (c,n) {\le 1}, \end{aligned}$$

while if $c>\overline{\delta }$, then

$$\begin{aligned} \lim _{n\rightarrow \infty } g (c,n)=0. \end{aligned}$$

Moreover, if the jump distribution has a finite right range R, then $\overline{\delta } < R$. If furthermore the jump distribution is “nearly right-continuous”, then there exists $\kappa \in (0,1]$ such that $\lim _{n\rightarrow \infty }g(c,n)=\kappa$ for all $c<\underline{\delta }$. We also show that the tail distribution of $M:=\sup _{n\ge 0}M_{n}$, namely, the rightmost position ever reached by the branching random walk, has a similar exponential decay (without the cutoff at $\underline{\delta }$). Finally, by duality, these results imply that the maximal displacement of supercritical branching random walks conditional on extinction has a similar tail behavior.