Mathematics – Probability
Scientific paper
2011-10-10
Mathematics
Probability
46 pages, 2 figures. v2: Exposition improved
Scientific paper
We are interested in the asymptotic behavior of critical Galton-Watson trees whose offspring distribution may have infinite variance, which are conditioned on having a large fixed number of leaves. We first find an asymptotic estimate of the probability for a critical Galton-Watson tree to have $n$ leaves. Secondly, we let $\t_n$ be a critical Galton-Watson tree whose offspring distribution is in the domain of attraction of a stable law, and conditioned on having exactly $n$ leaves. We show that the rescaled Lukasiewicz path and contour function of $\t_n$ converge respectively to $\X$ and $\H$, where $\X$ is the normalized excursion of a strictly stable spectrally positive L\'evy process and $\H$ is its associated continuous-time height function. As an application, we investigate the distribution of the maximum degree in a critical Galton-Watson tree conditioned on having a large number of leaves. We also explain how these results can be generalized to the case of Galton-Watson trees which are conditioned on having a large fixed number of vertices with degree in a given set, thus extending results obtained by Aldous, Duquesne and Rizzolo.
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