Necessary and sufficient conditions for the convergence of the consistent maximal displacement of the branching random walk (Q2415496)

Consider a supercritical branching random walk on the real line. Assume it to be in the ``boundary case'', see \textit{J. D. Biggins} and \textit{A. E. Kyprianou} [Electron. J. Probab. 10, Paper No. 17, 609--631 (2005; Zbl 1110.60081)]. Denote by \(V(u)\) the position of the individual \(u\), by \(|u|\) the generation \(u\) belongs to, and by \(u_k\) the ancestor of \(u\) alive at generation \(k\). The consistent maximal displacement is then defined by \(L_n :=\min_{|u|=n}\max_{k\le n}V(u_k)\). Suppose \(\sigma^2:= \mathbf{E} (\sum_{|u|=1} V(u)^2 e^{V(u)})<\infty\), and set \(\chi:= \sum_{|u|=1} e^{-V(u)}\). Using spinal decomposition, which links additive moments of the branching random walk with random walk estimates, and an expansion of results by \textit{A. A. Mogul'skij} [Theory Probab. Appl. 19, 726--736 (1974; Zbl 0326.60061); translation from Teor. Veroyatn. Primen. 19, 755--765 (1974)], on small deviations for random walk trajectories, the author proves that the assumption \(\lim_{x\to\infty}x^2\mathbf{E}(\sum_{|u| =1} e^{-V(u)}\mathbf{1}_{\log\chi\ge x})=0\) is necessary and sufficient for \(\lim_{x\to\infty}n^{-1/3}L_n= (3\pi^2 \sigma^2/2)^{1/3}\) almost surely on the event of survival. That is considerably stronger than what was previously known about the asymptotic growth of \(L_n\), see [\textit{M. Fang} and \textit{O. Zeitouni}, Electron. Commun. Probab. 15, 106--118 (2010; Zbl 1201.60041); \textit{G. Faraud} et al., Probab. Theory Relat. Fields 154, No. 3--4, 621--660 (2012; Zbl 1257.05162)].