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July 2000 The maximum of a branching random walk with semiexponential increments
Nina Gantert
Ann. Probab. 28(3): 1219-1229 (July 2000). DOI: 10.1214/aop/1019160332

Abstract

We consider an in .nite Galton–Watson tree $\Gamma$ and label the vertices $v$ with a collection of i.i.d.random variables $(Y_v)_{v \in \Gamma}$. In the case where the upper tail of the distribution of $Y_v$ is semiexponential, we then determine the speed of the corresponding tree-indexed random walk. In contrast to the classical case where the random variables $Y_v$ have finite exponential moments, the normalization in the definition of the speed depends on the distribution of $Y_v$. Interpreting the random variables $Y_v$ as displacements of the offspring from the parent, $(Y_v)_{v \in \Gamma}$ describes a branching random walk. The result on the speed gives a limit theorem for the maximum of the branching random walk, that is, for the position of the rightmost particle. In our case, this maximum grows faster than linear in time.

Citation

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Nina Gantert. "The maximum of a branching random walk with semiexponential increments." Ann. Probab. 28 (3) 1219 - 1229, July 2000. https://doi.org/10.1214/aop/1019160332

Information

Published: July 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1023.60073
MathSciNet: MR1797310
Digital Object Identifier: 10.1214/aop/1019160332

Subjects:
Primary: 60G50 , 60J15 , 60J80 , 60K35

Keywords: Branching random walk , Galton –Watson tree , semiexponential distributions , sums of i.i.d.random variables , tree-indexed random walk

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.28 • No. 3 • July 2000
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