On the maximal offspring in a subcritical branching process

Abstract

We consider a subcritical Galton–Watson tree $\mathsf {T}_{n}^{\Omega }$ conditioned on having $n$ vertices with outdegree in a fixed set $\Omega $. The offspring distribution is assumed to have a regularly varying density such that it lies in the domain of attraction of an $\alpha $-stable law for $1<\alpha \le 2$. Our main results consist of a local limit theorem for the maximal degree of $\mathsf {T}_{n}^{\Omega }$, and various limits describing the global shape of $\mathsf {T}_{n}^{\Omega }$. In particular, we describe the joint behaviour of the fringe subtrees dangling from the vertex with maximal degree. We provide applications of our main results to establish limits of graph parameters, such as the height, the non-maximal vertex outdegrees, and fringe subtree statistics.