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March 2013 Sub-Gaussian tail bounds for the width and height of conditioned Galton–Watson trees
Louigi Addario-Berry, Luc Devroye, Svante Janson
Ann. Probab. 41(2): 1072-1087 (March 2013). DOI: 10.1214/12-AOP758

Abstract

We study the height and width of a Galton–Watson tree with offspring distribution $\xi$ satisfying $\mathbb{E} \xi=1$, $0<\operatorname{Var} \xi<\infty$, conditioned on having exactly $n$ nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the width (largest number of nodes in any level) and height (greatest level containing a node); the bounds are optimal up to constant factors in the exponent. Under the same conditioning, we also derive essentially optimal upper tail bounds for the number of nodes at level $k$, for $1\leq k\leq n$.

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Louigi Addario-Berry. Luc Devroye. Svante Janson. "Sub-Gaussian tail bounds for the width and height of conditioned Galton–Watson trees." Ann. Probab. 41 (2) 1072 - 1087, March 2013. https://doi.org/10.1214/12-AOP758

Information

Published: March 2013
First available in Project Euclid: 8 March 2013

zbMATH: 1278.60128
MathSciNet: MR3077536
Digital Object Identifier: 10.1214/12-AOP758

Subjects:
Primary: 60C05 , 60J80

Keywords: Galton–Watson trees , Height , Random trees , Simply generated trees , width

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 2 • March 2013
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