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A Simple Proof of Duquesne’s Theorem on Contour Processes of Conditioned Galton–Watson Trees

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Séminaire de Probabilités XLV

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2078))

Abstract

We give a simple new proof of a theorem of Duquesne, stating that the properly rescaled contour function of a critical aperiodic Galton–Watson tree, whose offspring distribution is in the domain of attraction of a stable law of index θ ∈ (1, 2], conditioned on having total progeny n, converges in the functional sense to the normalized excursion of the continuous-time height function of a strictly stable spectrally positive Lévy process of index θ. To this end, we generalize an idea of Le Gall which consists in using an absolute continuity relation between the conditional probability of having total progeny exactly n and the conditional probability of having total progeny at least n. This new method is robust and can be adapted to establish invariance theorems for Galton–Watson trees having n vertices whose degrees are prescribed to belong to a fixed subset of the positive integers.

AMS 2000 subject classifications. Primary 60J80, 60F17, 60G52; secondary 05C05.

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Acknowledgements

I am deeply indebted to Jean-François Le Gall for insightful discussions and for making many useful suggestions on first versions of this manuscript.

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Authors and Affiliations

  1. Laboratoire de mathématiques, UMR 8628 CNRS, Université Paris-Sud, 91405, ORSAY Cedex, France

    Igor Kortchemski

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  1. Igor Kortchemski
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Correspondence to Igor Kortchemski .

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Editors and Affiliations

  1. , Laboratoire de Mathématiques, Université de Versailles-St Quentin, avenue des États-Unis 45, Versailles, 78035, France

    Catherine Donati-Martin

  2. IECN, Campus scientifique, BP 239, Nancy-Université, INRIA, Vandoeuvre-lès-Nancy, 54506, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St-Quentin, avenue des Etats-Unis 45, Versailles, 78035, France

    Alain Rouault

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Kortchemski, I. (2013). A Simple Proof of Duquesne’s Theorem on Contour Processes of Conditioned Galton–Watson Trees. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLV. Lecture Notes in Mathematics(), vol 2078. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00321-4_20

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