The maximum of a branching random walk with semiexponential increments

We consider an infinite Galton-Watson tree Gamma and label the vertices nu with a collection of i.i.d. random variables (Y-nu)(nu is an element of Gamma). In the case where the upper tail of the distribution of Y-nu is semiexponential, we then determine the speed of the corresponding tree-indexed random walk. In contrast to the classical case where the random variables Y-nu have finite exponential moments, the normalization in the definition of the speed depends on the distribution of Y-nu. Interpreting the random variables Y-nu as displacements of the offspring from the parent, (Y-nu)(nu is an element of Gamma) 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.