Asymptotics of the Hurwitz binomial distribution related to mixed Poisson Galton-Watson trees (Q2731581)

Hurwitz's extension of Abel's binomial theorem defines a probability distribution of a random subset \(V(n)\) of the set \([0,n]= \{0,1,\dots, n\}\). The random set \(V(n)\) can be constructed in a natural way as the set of vertices of a suitably defined random tree with \(n+2\) vertices, and the distribution on \([0,n]\) of the size of \(V(n)\) is proved to define a generalization of the binomial distribution. The main purpose of the paper is to describe the asymptotic behaviour of the \(V(n)\) distribution in a limiting regime in which the fringe subtree converges (in distribution) to a Galton-Watson tree, with a mixed Poisson offspring distribution.