A note on the maximal out-degree of Galton-Watson trees

TL;DR

This paper investigates the behavior of the maximum number of offspring in Galton-Watson trees, revealing asymptotic relationships between local and global maxima and offspring distribution tail behavior.

Contribution

It establishes the asymptotic equivalence of local maximal out-degree tail and offspring distribution tail, and clarifies conditions under which this holds for global maxima.

Findings

Local maximal out-degree tail matches offspring distribution tail asymptotically.

Global maximal out-degree tail matches offspring distribution tail only in subcritical cases.

Provides insights into the extremal structure of Galton-Watson trees.

Abstract

In this note we consider both the local maximal out-degree and the global maximal out-degree of Galton-Watson trees. In particular, we show that the tail of any local maximal out-degree and that of the offspring distribution are asymptotically of the same order. However for the global maximal out-degree, this is only true in the subcritical case.