A simple proof of Duquesne's theorem on contour processes of conditioned Galton-Watson trees

Details A simple proof of Duquesne's theorem on contour processes of conditioned Galton-Watson trees A simple proof of Duquesne's theorem on contour processes of conditioned Galton-Watson trees

2011-09-19

arxiv.org/abs/1109.4138v1

Mathematics

Probability

17 pages, 2 figures

Scientific paper

We give a new simple 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 $\theta \in (1,2]$, 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 L\'evy process of index $\theta$. 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.

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