The Continuum Random Tree III

David Aldous
1993 Annals of Probability  
Let (W(k), k 2 1) be random trees with k leaves, satisfying a consistency condition: Removing a random leaf from R(k) gives R(k -1). Then under an extra condition, this family determines a random continuum tree ?/, which it is convenient to represent as a random subset of 11. This leads to an abstract notion of convergence in distribution, as n -A o, of (rescaled) random trees En-on n vertices to a limit continuum random tree I/. The notion is based upon the assumption that, for fixed k, the
more » ... trees of En determined by k randomly chosen vertices converge to R(k). As our main example, under mild conditions on the offspring distribution, the family tree of a Galton-Watson branching process, conditioned on total population size equal to n, can be rescaled to converge to a limit continuum random tree which can be constructed from Brownian excursion.
doi:10.1214/aop/1176989404 fatcat:4oyqnmkw5nfbvivyfuipdchaie