Abstract
This work proves new probability bounds relating to the height, width, and size of Galton–Watson trees. For example, if T is any Galton–Watson tree, and H, W, and |T| are the height, width, and size of T, respectively, then H / W has sub-exponential tails and \(H/|T|^{1/2}\) has sub-Gaussian tails. Although our methods apply without any assumptions on the offspring distribution, when information is provided about the distribution the method can be adapted accordingly, and always seems to yield tight bounds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Addario-Berry, L., Devroye, L., Janson, S.: Sub-Gaussian tail bounds for the width and height of conditioned Galton-Watson trees. Ann. Probab. 41(2), 1072–1087 (2013). https://doi.org/10.1214/12-AOP758
Duquesne, T., Wang, M.: Decomposition of Lévy trees along their diameter. Ann. Inst. H. Poincaré Probab. Stat. 53(2), 539–593 (2017). https://doi.org/10.1214/15-AIHP725. https://arxiv.org/abs/1503.05069
Esseen, C.G.: On the concentration function of a sum of independent random variables. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9, 290–308 (1968). https://doi.org/10.1007/BF00531753
Haas, B.: Loss of mass in deterministic and random fragmentations. Stoch. Process. Appl. 106(2):245–277 (2003). ISSN 0304-4149. https://doi.org/10.1016/S0304-4149(03)00045-0. http://www.proba.jussieu.fr/mathdoc/textes/PMA-730.pdf
Janson, S.: Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation. Probab. Surv. 9, 103–252 (2012). ISSN 1549-5787. https://doi.org/10.1214/11-PS188
Kesten, K.:. Sums of independent random variables—without moment conditions. Ann. Math. Stat. 43, 701–732 (1972). ISSN 0003-4851. https://doi.org/10.1214/aoms/1177692541
Kolmogorov, A.N.: Sur les propriétés des fonctions de concentrations de M.P. Lévy. Ann. Inst. H. Poincaré 16(1), 27–34 (1958). http://www.numdam.org/article/AIHP_1958__16_1_27_0.pdf
Kortchemski, I.: Sub-exponential tail bounds for conditioned stable Bienaymé–Galton–Watson trees. Probab. Theory Relat. Fields (to appear) 160(1–2), 1–40 (2017). https://doi.org/10.1007/s00440-016-0704-6. https://arxiv.org/abs/1504.04358
Le Jean-François, G.: Random trees and applications. Probab. Surv. 2, 245–311 (2005). https://doi.org/10.1214/154957805100000140. https://projecteuclid.org/euclid.ps/1132583290
Lévy, P.: Théorie de l’addition des Variables aléatoires, 2nd edn. Gauthier-Villars, Paris (1954)
Acknowledgements
Thanks to Igor Kortchemski and to Yuval Peres, for useful discussions during the preparation of this paper. Thanks also to an anonymous referee for an extremely careful and thorough reading of the paper and for many helpful remarks. This research was funded in part by an NSERC Discovery Grant. Part of the research was carried out at the Isaac Newton Institute for Mathematical Sciences during the programme “Random Geometry”, supported by EPSRC Grant No. EP/K032208/1. My interest in this problem was sparked during a workshop at McGill’s Bellairs research institute in Holetown, Barbados.
