The Annals of Probability

Sub-Gaussian tail bounds for the width and height of conditioned Galton–Watson trees

Louigi Addario-Berry, Luc Devroye, and Svante Janson

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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 $1\leq k\leq n$.

Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 1072-1087.

Dates
First available in Project Euclid: 8 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1362750951

Digital Object Identifier
doi:10.1214/12-AOP758

Mathematical Reviews number (MathSciNet)
MR3077536

Zentralblatt MATH identifier
1278.60128

Subjects
Primary: 60C05: Combinatorial probability 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Random trees Galton–Watson trees simply generated trees width height

Citation

Addario-Berry, Louigi; Devroye, Luc; Janson, Svante. Sub-Gaussian tail bounds for the width and height of conditioned Galton–Watson trees. Ann. Probab. 41 (2013), no. 2, 1072--1087. doi:10.1214/12-AOP758. https://projecteuclid.org/euclid.aop/1362750951


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