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2020 Large deviations for the maximum of a branching random walk with stretched exponential tails
Piotr Dyszewski, Nina Gantert, Thomas Höfelsauer
Electron. Commun. Probab. 25: 1-13 (2020). DOI: 10.1214/20-ECP353

Abstract

We prove large deviation results for the position of the rightmost particle, denoted by $M_{n}$, in a one-dimensional branching random walk in a case when Cramér’s condition is not satisfied. More precisely we consider step size distributions with stretched exponential upper and lower tails, i.e. both tails decay as $e^{-\Theta (|t|^{r})}$ for some $r\in ( 0,1)$. It is known that in this case, $M_{n}$ grows as $n^{1/r}$ and in particular faster than linearly in $n$. Our main result is a large deviation principle for the laws of $n^{-1/r}M_{n}$ . In the proof we use a comparison with the maximum of (a random number of) independent random walks, denoted by $\tilde{M} _{n}$, and we show a large deviation principle for the laws of $n^{-1/r}\tilde{M} _{n}$ as well.

Citation

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Piotr Dyszewski. Nina Gantert. Thomas Höfelsauer. "Large deviations for the maximum of a branching random walk with stretched exponential tails." Electron. Commun. Probab. 25 1 - 13, 2020. https://doi.org/10.1214/20-ECP353

Information

Received: 15 June 2020; Accepted: 25 September 2020; Published: 2020
First available in Project Euclid: 2 October 2020

Digital Object Identifier: 10.1214/20-ECP353

Subjects:
Primary: 60F10, 60G50, 60J80

JOURNAL ARTICLE
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