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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


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.


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Nina Gantert. "The maximum of a branching random walk with semiexponential increments." Ann. Probab. 28 (3) 1219 - 1229, July 2000.


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

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

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


Vol.28 • No. 3 • July 2000
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