Invariance principles for Galton-Watson trees conditioned on the number of leaves

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:122

Article

Kortchemski, Igor

关键词: Random trees;    Invariance principles;    Scaling limits;    Conditioned Gallon-Watson trees;    Stable trees;   

DOI  :  10.1016/j.spa.2012.05.013

来源: Elsevier

【 摘 要 】

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 for the probability of a Gallon-Watson tree having n leaves. Second, 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 tn converge respectively to X-exc and H-exc where X-exc is the normalized excursion of a strictly stable spectrally positive Levy process and H-exc 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. (C) 2012 Elsevier B.V. All rights reserved.

【 授权许可】

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