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On large-deviation probabilities for the empirical distribution of branching random walks with heavy tails

Part of: Stochastic processes Markov processes Limit theorems

Published online by Cambridge University Press:  24 March 2022

Shuxiong Zhang
Shuxiong Zhang*
Affiliation:
Beijing Normal University
*
*Postal address: School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China. Email: shuxiong.zhang@qq.com
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Abstract

Given a branching random walk $(Z_n)_{n\geq0}$ on $\mathbb{R}$ , let $Z_n(A)$ be the number of particles located in interval A at generation n. It is well known that under some mild conditions, $Z_n(\sqrt nA)/Z_n(\mathbb{R})$ converges almost surely to $\nu(A)$ as $n\rightarrow\infty$ , where $\nu$ is the standard Gaussian measure. We investigate its large-deviation probabilities under the condition that the step size or offspring law has a heavy tail, i.e. a decay rate of $\mathbb{P}(Z_n(\sqrt nA)/Z_n(\mathbb{R})>p)$ as $n\rightarrow\infty$ , where $p\in(\nu(A),1)$ . Our results complete those in Chen and He (2019) and Louidor and Perkins (2015).

Keywords

Step sizeoffspring lawSchröder constantCramér’s theorem

MSC classification

Primary: 60F10: Large deviations
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G50: Sums of independent random variables; random walks
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 2 , June 2022 , pp. 471 - 494
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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