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 \[ g(c,n):=ρ^{cn} P(M_{n}\geq cn), \quad \mbox{for each }c>0 \mbox{ and } n\in\mathbb{N}, \] satisfies the following properties: there exist $0<\underline{\delta }\le \overline{\delta }<\infty$ such that if $c<\underline{\delta }$, then $$0<\underset{n\to \infty }{\mathrm{liminf}}g(c,n)\le \underset{n\to \infty }{\mathrm{limsup}}g(c,n)\le 1,$$ while if $c>\overline{\delta }$, then \[ \lim_{n\rightarrow\infty} g (c,n)=0. \] 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 ${\lim }_{n\to \infty }g(c,n)=\kappa$ 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.

29 pages