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January, 1993 The Continuum Random Tree III
David Aldous
Ann. Probab. 21(1): 248-289 (January, 1993). DOI: 10.1214/aop/1176989404

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.

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David Aldous. "The Continuum Random Tree III." Ann. Probab. 21 (1) 248 - 289, January, 1993. https://doi.org/10.1214/aop/1176989404

Information

Published: January, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0791.60009
MathSciNet: MR1207226
Digital Object Identifier: 10.1214/aop/1176989404

Subjects:
Primary: 60C05
Secondary: 60B10 , 60J80

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

Rights: Copyright © 1993 Institute of Mathematical Statistics

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Vol.21 • No. 1 • January, 1993
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