The maximum of a tree-indexed random walk in the big jump domain

TL;DR

This paper establishes a criterion for when the maximum displacement in a tree-indexed random walk is dominated by a single large jump, with applications to critical Galton--Watson trees in the stable law domain.

Contribution

It introduces a simple criterion based on tail behavior, expectation, tree height, and size to determine the maximum displacement in tree-indexed random walks.

Findings

Maximum displacement often caused by a single large jump under the criterion.

Criterion applies to various trees, including critical Galton--Watson trees.

Results extend understanding of extremal behavior in tree-indexed processes.

Abstract

We consider random walks indexed by arbitrary finite random or deterministic trees. We derive a simple sufficient criterion which ensures that the maximal displacement of the tree-indexed random walk is determined by a single large jump. This criterion is given in terms of four quantities : the tail and the expectation of the random walk steps, the height of the tree and the number of its vertices. The results are applied to critical Galton--Watson trees with offspring distributions in the domain of attraction of a stable law.