On the maximal displacement of subcritical branching random walks

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 satisfies the following properties: there exist \(0<\underline{\delta }\le \overline{\delta } < {\infty }\) such that if \(c<\underline{\delta }\), then while if \(c>\overline{\delta }\), then 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.