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

Bastien Mallein (1)

1 DMA - Département de Mathématiques et Applications - ENS Paris

We consider a supercritical branching random walk on R. The consistent maximal displacement is the smallest of the distances between the trajectories of individuals at the nth generation and the boundary of the process. It has been proved by Fang and Zeitouni [7] and by Faraud, Hu and Shi [8] that the consistent maximal displacement grows almost surely at rate λ * n 1/3 for an explicit λ *. We obtain here a necessary and sufficient condition for this asymptotic behaviour to hold.

Mots clés: branching random walk; random walk; small deviations

Domaines: Probabilités [math.PR]

Bastien Mallein. Necessary and sufficient conditions for the convergence of the consistent maximal displacement of the branching random walk. Brazilian Journal of Probability and Statistics, 2019, 33 (2). ⟨hal-01322467⟩