Asymptotics of the Hurwitz Binomial Distribution Related to Mixed Poisson Galton–Watson Trees

Authors:
Jürgen Bennies;Jim Pitman
Affiliations:
Department of Statistics, University of California, 367 Evans Hall # 3860, Berkeley, CA 94720-3860, USA (e-mail: pitman@stat.berkeley.edu);Department of Statistics, University of California, 367 Evans Hall # 3860, Berkeley, CA 94720-3860, USA (e-mail: pitman@stat.berkeley.edu)
Venue:
Combinatorics, Probability and Computing
Year:
2001

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Abstract

Hurwitz's extension of Abel's binomial theorem defines a probability distribution on the set of integers from 0 to n. This is the distribution of the number of non-root vertices of a fringe subtree of a suitably defined random tree with n + 2 vertices. The asymptotic behaviour of this distribution is described in a limiting regime in which the fringe subtree converges in distribution to a Galton–Watson tree with a mixed Poisson offspring distribution.