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Itô's excursion theory and random trees

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  • Le Gall, Jean-François

Abstract

We explain how Itô's excursion theory can be used to understand the asymptotic behavior of large random trees. We provide precise statements showing that the rescaled contour of a large Galton-Watson tree is asymptotically distributed according to Itô's excursion measure. As an application, we provide a simple derivation of Aldous' theorem stating that the rescaled contour function of a Galton-Watson tree conditioned to have a fixed large progeny converges to a normalized Brownian excursion. We also establish a similar result for a Galton-Watson tree conditioned to have a fixed large height.

Suggested Citation

  • Le Gall, Jean-François, 2010. "Itô's excursion theory and random trees," Stochastic Processes and their Applications, Elsevier, vol. 120(5), pages 721-749, May.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:5:p:721-749
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    References listed on IDEAS

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    1. Watanabe, Shinzo, 2010. "Itô's theory of excursion point processes and its developments," Stochastic Processes and their Applications, Elsevier, vol. 120(5), pages 653-677, May.
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    Cited by:

    1. Normand, Raoul, 2014. "Two population models with constrained migrations," Stochastic Processes and their Applications, Elsevier, vol. 124(5), pages 1773-1812.
    2. Bertoin, Jean, 2010. "A limit theorem for trees of alleles in branching processes with rare neutral mutations," Stochastic Processes and their Applications, Elsevier, vol. 120(5), pages 678-697, May.

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    2. Bertoin, Jean, 2010. "A limit theorem for trees of alleles in branching processes with rare neutral mutations," Stochastic Processes and their Applications, Elsevier, vol. 120(5), pages 678-697, May.

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