Biased branching random walks on Bienaymé--Galton--Watson trees

We study $\lambda$-biased branching random walks on Bienaymé–Galton–Watson trees in discrete time. We consider the maximal displacement at time $n$, ${\max }_{\mathrm{\mid }u\mathrm{\mid }=n}\mathrm{\mid }X(u)\mathrm{\mid }$, and show that it almost surely grows at a deterministic, linear speed. We characterize this speed with the help of the large deviation rate function of the $\lambda$-biased random walk of a single particle. A similar result is given for the minimal displacement at time $n$, ${\min }_{\mathrm{\mid }u\mathrm{\mid }=n}\mathrm{\mid }X(u)\mathrm{\mid }$.