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2020 On the maximal offspring in a subcritical branching process
Benedikt Stufler
Electron. J. Probab. 25: 1-62 (2020). DOI: 10.1214/20-EJP506


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.


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Benedikt Stufler. "On the maximal offspring in a subcritical branching process." Electron. J. Probab. 25 1 - 62, 2020.


Received: 17 January 2019; Accepted: 3 August 2020; Published: 2020
First available in Project Euclid: 4 September 2020

zbMATH: 07252698
Digital Object Identifier: 10.1214/20-EJP506

Primary: 60F17 , 60J80
Secondary: 05C0 , 05C80

Keywords: condensation phenomena , limits of graph parameters , Random trees


Vol.25 • 2020
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