Stufler, Benedikt
(2020):
On the maximal offspring in a subcritical branching process.
In: Electronic Journal of Probability, Vol. 25, 104

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Abstract
We consider a subcritical GaltonWatson tree Tn(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 alphastable law for 1 <= alpha <= 2. Our main results consist of a local limit theorem for the maximal degree of Tn(Omega) , and various limits describing the global shape of Tn(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 nonmaximal vertex outdegrees, and fringe subtree statistics.