The maximum of a branching random walk with semiexponential increments

Gantert, Nina

Ann. Probab., Tome 28 (2000) no. 1, p. 1219-1229 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2000-06-14
Classification:  Branching random walk,  tree-indexed random walk,  Galton –Watson tree,  semiexponential distributions,  sums of i.i.d.random variables,  60J80,  60J15,  60G50,  60K35