The almost sure limits of the minimal position and the additive martingale in a branching random walk

Let {V (u), u ∈ T} be a discrete-time branching random walk on the real line R, where T is an Ulam-Harris tree which describes the genealogy of the particles and V (u) ∈ R is the position of the particle u. When a particle u is at n-th generation, we write |u| = n for n ≥ 0. The branching random walk V can be described as follows: At the beginning, there is a single particle ∅ located at 0. The particle ∅ is also the root of T. At the generation 1, the root dies and gives birth to some point process L on R. The point process L constitutes the first generation of the branching random walk {V (u), |u| = 1}. The next generations are defined by recurrence: For each |u| = n (if such u exists), the particle u dies at the (n + 1)-th generation and gives birth to an independent copy of L shifted by V (u). The collection of all children of all u together with their positions gives the (n + 1)-th generation. The whole system may survive forever or die out after some generations. Plainly L = |u|=1 δ{V (u)}. Assume E[L(R)] > 1 and that