The maximum of a branching random walk with semiexponential increments

Abstract

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