Note on the Generalized Branching Random Walk on the Galton–Watson Tree

Let (Formula presented.) be a super-critical Galton–Watson tree. Recently, the first author computed almost surely and simultaneously the Hausdorff dimensions of the sets of infinite branches of the boundary of (Formula presented.) along which the sequence (Formula presented.) has a given set of limit points, where (Formula presented.) and (Formula presented.) are two branching random walks defined on (Formula presented.). In this study, we are interested in the study of the speed of convergence of this sequence. More precisely, for a given sequence (Formula presented.), we consider (Formula presented.) We will give a sufficient condition on (Formula presented.) so that (Formula presented.) has a maximal Hausdorff and packing dimension.