The continuum random tree. III. (English) Zbl 0791.60009

Aldous, David

Ann. Probab. 21, No. 1, 248-289 (1993).

Summary: [For parts I and II see author, ibid. 19, No. 1, 1-28 (1991; Zbl 0722.60013) and the paper reviewed above.]
Let \(({\mathcal R} (k)\), \(k \geq 1)\) be random trees with \(k\) leaves, satisfying a consistency condition: Removing a random leaf from \({\mathcal R} (k)\) gives \({\mathcal R} (k-1)\). Then under an extra condition, this family determines a random continuum tree \({\mathcal S}\), which it is convenient to represent as a random subset of \(l_ 1\). This leads to an abstract notion of convergence in distribution, as \(n \to \infty\), of (rescaled) random trees \({\mathcal T}_ n\) on \(n\) vertices to a limit continuum random tree \({\mathcal S}\). The notion is based upon the assumption that, for fixed \(k\), the subtrees of \({\mathcal T}_ n\) determined by \(k\) randomly chosen vertices converge to \({\mathcal R} (k)\). 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.

MSC:

60C05	Combinatorial probability
60B10	Convergence of probability measures
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
05C05	Trees
05C80	Random graphs (graph-theoretic aspects)

Keywords:

weak convergence; random trees; Galton-Watson branching process; Brownian excursion

Citations:

Zbl 0791.60008; Zbl 0722.60013