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Consider a supercritical branching random walk on the real line. Theconsistent maximal displacement is the smallest of the distances between thetrajectories followed by individuals at the $n$th generation and the boundaryof the process. Fang and Zeitouni, and Faraud, Hu and Shi proved that undersome integrability conditions, the consistent maximal displacement grows almostsurely at rate $\lambda^* n^{1/3}$ for some explicit constant $\lambda^*$. Weobtain here a necessary and sufficient condition for this asymptotic behaviourto hold.

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