The Annals of Probability

The Continuum Random Tree III

David Aldous

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Abstract

Let $(\mathscr{R}(k), k \geq 1)$ be random trees with $k$ leaves, satisfying a consistency condition: Removing a random leaf from $\mathscr{R}(k)$ gives $\mathscr{R}(k - 1)$. Then under an extra condition, this family determines a random continuum tree $\mathscr{L}$, which it is convenient to represent as a random subset of $l_1$. This leads to an abstract notion of convergence in distribution, as $n \rightarrow \infty$, of (rescaled) random trees $\mathscr{J}_n$ on $n$ vertices to a limit continuum random tree $\mathscr{L}$. The notion is based upon the assumption that, for fixed $k$, the subtrees of $\mathscr{J}_n$ determined by $k$ randomly chosen vertices converge to $\mathscr{R}(k)$. 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.

Article information

Source
Ann. Probab. Volume 21, Number 1 (1993), 248-289.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989404

Digital Object Identifier
doi:10.1214/aop/1176989404

Mathematical Reviews number (MathSciNet)
MR1207226

Zentralblatt MATH identifier
0791.60009

JSTOR
links.jstor.org

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 60B10: Convergence of probability measures 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Random tree Galton-Watson branching process Brownian excursion weak convergence

Citation

Aldous, David. The Continuum Random Tree III. Ann. Probab. 21 (1993), no. 1, 248--289. doi:10.1214/aop/1176989404. https://projecteuclid.org/euclid.aop/1176989404


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