The maximum of a branching random walk with semiexponential increments

We consider an in .nite Galton–Watson tree Γ and label the vertices v with a collection of i.i.d.random variables (Yᵥ)v ∈ Γ. In the case where the upper tail of the distribution of Yᵥ is semiexponential, we then determine the speed of the corresponding tree-indexed random walk. In contrast to the classical case where the random variables Yᵥ have finite exponential moments, the normalization in the definition of the speed depends on the distribution of Yᵥ. Interpreting the random variables Yᵥ as displacements of the offspring from the parent, (Yᵥ)v ∈ Γ describes a branching random walk. The result on the speed gives a limit theorem for the maximum of the branching random walk, that is, for the position of the rightmost particle. In our case, this maximum grows faster than linear in time.